For the complete documentation index, see llms.txt.

Double Integral Calculator API

math 1 credit / call v2026-04-22

Evaluate an iterated double integral over rectangular or variable-limit regions.

One POST adds Double Integral Calculator to your app, site, workflow, or agent — formula, validation, edge cases, and docs already handled.

prefer a UI? Open the Double Integral Calculator on miniwebtool.com →

Start free — get a key → 1,000 credits · no card · 30 seconds

Endpoint

POST · 1cr
POST https://api.miniwebtool.com/v1/tools/double-integral-calculator/run

Request body

Field Type Req.
function
integrand
 str —
outer_variable
single letter, integrated last
 str —
inner_variable
single letter, integrated first
 str —
outer_lower
constant lower limit
 str —
outer_upper
constant upper limit
 str —
inner_lower
may depend on the outer variable
 str —
inner_upper
may depend on the outer variable
 str —

Cost & access

  • 1 credit per successful call — failed calls refund.
  • Starts on: Free.
  • Max payload: 65536 bytes.
  • Privacy mode: hash_only
  • Available on: free, starter, pro, business, scale

Response envelope

{
  "request_id": "01K...",
  "tool": "double-integral-calculator",
  "tool_version": "2026-04-22",
  "credits_used": 1,
  "result": { ... }
}

Try it

Stored only in this browser tab. Playground test key is prefilled and only works on this page. Get a live key →
integrand
single letter, integrated last
single letter, integrated first
constant lower limit
constant upper limit
may depend on the outer variable
may depend on the outer variable

Get your own key

1,000 free calls/mo · no card · key arrives by email

One verification click, then your key + a ready-to-run request. No card, no spam.

Code examples

curl -X POST https://api.miniwebtool.com/v1/tools/double-integral-calculator/run \
  -H 'Authorization: Bearer mwt_live_YOUR_KEY' \
  -H 'Content-Type: application/json' \
  -d '{"function":"x*y","outer_variable":"x","inner_variable":"y","outer_lower":"0","outer_upper":"1","inner_lower":"0","inner_upper":"1"}'
import requests

resp = requests.post(
    'https://api.miniwebtool.com/v1/tools/double-integral-calculator/run',
    headers={'Authorization': 'Bearer mwt_live_YOUR_KEY'},
    json={'function': 'x*y', 'outer_variable': 'x', 'inner_variable': 'y', 'outer_lower': '0', 'outer_upper': '1', 'inner_lower': '0', 'inner_upper': '1'},
)
resp.raise_for_status()
data = resp.json()
print(data)
const resp = await fetch(
  'https://api.miniwebtool.com/v1/tools/double-integral-calculator/run',
  {
    method: 'POST',
    headers: {
      'Authorization': 'Bearer mwt_live_YOUR_KEY',
      'Content-Type': 'application/json',
    },
    body: JSON.stringify({"function":"x*y","outer_variable":"x","inner_variable":"y","outer_lower":"0","outer_upper":"1","inner_lower":"0","inner_upper":"1"}),
  }
);
const data = await resp.json();
console.log(data);
<?php
$ch = curl_init('https://api.miniwebtool.com/v1/tools/double-integral-calculator/run');
curl_setopt_array($ch, [
    CURLOPT_POST => true,
    CURLOPT_RETURNTRANSFER => true,
    CURLOPT_HTTPHEADER => [
        'Authorization: Bearer mwt_live_YOUR_KEY',
        'Content-Type: application/json',
    ],
    CURLOPT_POSTFIELDS => json_encode([
    'function' => 'x*y',
    'outer_variable' => 'x',
    'inner_variable' => 'y',
    'outer_lower' => '0',
    'outer_upper' => '1',
    'inner_lower' => '0',
    'inner_upper' => '1'
]),
]);
$resp = curl_exec($ch);
curl_close($ch);
print_r(json_decode($resp, true));
require 'net/http'
require 'json'
require 'uri'

uri = URI('https://api.miniwebtool.com/v1/tools/double-integral-calculator/run')
http = Net::HTTP.new(uri.host, uri.port)
http.use_ssl = (uri.scheme == 'https')

req = Net::HTTP::Post.new(uri)
req['Authorization'] = 'Bearer mwt_live_YOUR_KEY'
req['Content-Type'] = 'application/json'
req.body = {
  'function' => 'x*y',
  'outer_variable' => 'x',
  'inner_variable' => 'y',
  'outer_lower' => '0',
  'outer_upper' => '1',
  'inner_lower' => '0',
  'inner_upper' => '1'
}.to_json

resp = http.request(req)
puts JSON.parse(resp.body).inspect
package main

import (
  "bytes"
  "encoding/json"
  "fmt"
  "io"
  "net/http"
)

func main() {
  body, _ := json.Marshal(map[string]interface{}{
    "function": "x*y",
    "outer_variable": "x",
    "inner_variable": "y",
    "outer_lower": "0",
    "outer_upper": "1",
    "inner_lower": "0",
    "inner_upper": "1",
  })
  req, _ := http.NewRequest("POST", "https://api.miniwebtool.com/v1/tools/double-integral-calculator/run", bytes.NewReader(body))
  req.Header.Set("Authorization", "Bearer mwt_live_YOUR_KEY")
  req.Header.Set("Content-Type", "application/json")

  resp, err := http.DefaultClient.Do(req)
  if err != nil { panic(err) }
  defer resp.Body.Close()
  out, _ := io.ReadAll(resp.Body)
  fmt.Println(string(out))
}
import java.net.URI;
import java.net.http.*;

var client = HttpClient.newHttpClient();
var req = HttpRequest.newBuilder()
    .uri(URI.create("https://api.miniwebtool.com/v1/tools/double-integral-calculator/run"))
    .header("Authorization", "Bearer mwt_live_YOUR_KEY")
    .header("Content-Type", "application/json")
    .POST(HttpRequest.BodyPublishers.ofString("{\"function\":\"x*y\",\"outer_variable\":\"x\",\"inner_variable\":\"y\",\"outer_lower\":\"0\",\"outer_upper\":\"1\",\"inner_lower\":\"0\",\"inner_upper\":\"1\"}"))
    .build();

var resp = client.send(req, HttpResponse.BodyHandlers.ofString());
System.out.println(resp.body());
agents let your coding agent do it

Copy this into Claude Code, Cursor, Codex, Aider or Cline. It carries the endpoint, the full schema, a sample response, and the credit rules — so the agent wires double-integral-calculator up correctly without draining your key.

Free to iterate: the prompt points the agent at POST /v1/tools/double-integral-calculator/dry-run, which validates the request and returns the response shape for 0 credits. Only the final confirmation call is billed (1 credit).
Stable URL for this prompt: /tools/double-integral-calculator/prompt.md (?variant=cursor / ?variant=mcp).
Paste into Claude Code, or save as CLAUDE.md context.
Integrate the MiniWebtool Double Integral Calculator API into this project.

## Your task

1. Read the reference below.
2. Ask me where this belongs in the codebase if it is not obvious.
3. Write the integration in the language and style of the surrounding code:
   a small typed client function, errors handled, key read from the
   environment (never hard-coded, never committed).
4. Verify with `/dry-run` first, then exactly one live call.
5. Show me the diff and the live result.

## Credit budget — follow these exactly

This API is metered. A live call costs 1 credit; the free tier
includes 1,000 credits per month. Burning them during integration is the
single most common way to waste this key.

1. **Do all wiring against `/dry-run`, which costs 0 credits.** It validates the
   payload for real and returns a sample result with the exact response shape.
   Iterate there until your request builds and your parsing works.
2. **Make at most ONE live `/run` call** — a single end-to-end confirmation once
   dry-run passes. Print the result, then stop.
3. **Never call the API from unit tests, examples, or a retry loop.** Assert
   against the sample response captured from `/dry-run` instead.
4. **On 4xx, fix the payload — do not retry.** The error body is RFC 7807
   `application/problem+json` and says exactly what is wrong.
5. **On 429, honour `Retry-After`** and back off; do not tighten the loop.
6. **Read `X-MWT-Credits-Remaining`** on every response. If it drops below 50,
   stop making live calls and tell me.
7. If the integration needs repeated calls at runtime, **cache by input** — this
   tool is deterministic, so the same input always returns the same output.

## The API

**Double Integral Calculator** — Evaluate an iterated double integral over rectangular or variable-limit regions.

- Live endpoint: `POST https://api.miniwebtool.com/v1/tools/double-integral-calculator/run` — costs 1 credit
- Dry run: `POST https://api.miniwebtool.com/v1/tools/double-integral-calculator/dry-run` — costs 0 credits, same auth and validation
- Auth: `Authorization: Bearer <MINIWEBTOOL_API_KEY>`
- Content type: `application/json`
- Tool version: `2026-04-22` (output shape is stable within a major version)
- Full machine-readable spec: `https://api.miniwebtool.com/v1/openapi.json`

### Request body

| field | type | required | notes |
|---|---|---|---|
| `function` | str | no | integrand (default `x*y`) |
| `outer_variable` | str | no | single letter, integrated last (default `x`) |
| `inner_variable` | str | no | single letter, integrated first (default `y`) |
| `outer_lower` | str | no | constant lower limit (default `0`) |
| `outer_upper` | str | no | constant upper limit (default `1`) |
| `inner_lower` | str | no | may depend on the outer variable (default `0`) |
| `inner_upper` | str | no | may depend on the outer variable (default `1`) |

Example request body:

```json
{
  "function": "x*y",
  "outer_variable": "x",
  "inner_variable": "y",
  "outer_lower": "0",
  "outer_upper": "1",
  "inner_lower": "0",
  "inner_upper": "1"
}
```

### Response envelope

```json
{
  "request_id": "req_01H…",
  "tool": "double-integral-calculator",
  "tool_version": "2026-04-22",
  "credits_used": 1,
  "result": {
    "integrand": "x*y",
    "iterated_integral": "integral(0 to 1) integral(0 to 1) x*y dy dx",
    "outer_variable": "x",
    "inner_variable": "y",
    "outer_limits": [
      0.0,
      1.0
    ],
    "inner_limits": [
      "0",
      "1"
    ],
    "region_type": "rectangular",
    "value": 0.25,
    "estimated_error": 0.0,
    "average_value": 0.25,
    "method": "composite Simpson with Richardson refinement"
  }
}
```

`result` holds the tool output. Errors come back as
`application/problem+json` with `type`, `title`, `status`, and `detail`.

### Getting a key

If `MINIWEBTOOL_API_KEY` is not already in the environment, stop and ask me for
one — do not sign up, scrape, or guess a key. Free keys: https://api.miniwebtool.com/dashboard/
Paste into Codex, Aider, Cline, or any coding agent.
Integrate the MiniWebtool Double Integral Calculator API into this project.

## Your task

1. Read the reference below.
2. Ask me where this belongs in the codebase if it is not obvious.
3. Write the integration in the language and style of the surrounding code:
   a small typed client function, errors handled, key read from the
   environment (never hard-coded, never committed).
4. Verify with `/dry-run` first, then exactly one live call.
5. Show me the diff and the live result.

## Credit budget — follow these exactly

This API is metered. A live call costs 1 credit; the free tier
includes 1,000 credits per month. Burning them during integration is the
single most common way to waste this key.

1. **Do all wiring against `/dry-run`, which costs 0 credits.** It validates the
   payload for real and returns a sample result with the exact response shape.
   Iterate there until your request builds and your parsing works.
2. **Make at most ONE live `/run` call** — a single end-to-end confirmation once
   dry-run passes. Print the result, then stop.
3. **Never call the API from unit tests, examples, or a retry loop.** Assert
   against the sample response captured from `/dry-run` instead.
4. **On 4xx, fix the payload — do not retry.** The error body is RFC 7807
   `application/problem+json` and says exactly what is wrong.
5. **On 429, honour `Retry-After`** and back off; do not tighten the loop.
6. **Read `X-MWT-Credits-Remaining`** on every response. If it drops below 50,
   stop making live calls and tell me.
7. If the integration needs repeated calls at runtime, **cache by input** — this
   tool is deterministic, so the same input always returns the same output.

## The API

**Double Integral Calculator** — Evaluate an iterated double integral over rectangular or variable-limit regions.

- Live endpoint: `POST https://api.miniwebtool.com/v1/tools/double-integral-calculator/run` — costs 1 credit
- Dry run: `POST https://api.miniwebtool.com/v1/tools/double-integral-calculator/dry-run` — costs 0 credits, same auth and validation
- Auth: `Authorization: Bearer <MINIWEBTOOL_API_KEY>`
- Content type: `application/json`
- Tool version: `2026-04-22` (output shape is stable within a major version)
- Full machine-readable spec: `https://api.miniwebtool.com/v1/openapi.json`

### Request body

| field | type | required | notes |
|---|---|---|---|
| `function` | str | no | integrand (default `x*y`) |
| `outer_variable` | str | no | single letter, integrated last (default `x`) |
| `inner_variable` | str | no | single letter, integrated first (default `y`) |
| `outer_lower` | str | no | constant lower limit (default `0`) |
| `outer_upper` | str | no | constant upper limit (default `1`) |
| `inner_lower` | str | no | may depend on the outer variable (default `0`) |
| `inner_upper` | str | no | may depend on the outer variable (default `1`) |

Example request body:

```json
{
  "function": "x*y",
  "outer_variable": "x",
  "inner_variable": "y",
  "outer_lower": "0",
  "outer_upper": "1",
  "inner_lower": "0",
  "inner_upper": "1"
}
```

### Response envelope

```json
{
  "request_id": "req_01H…",
  "tool": "double-integral-calculator",
  "tool_version": "2026-04-22",
  "credits_used": 1,
  "result": {
    "integrand": "x*y",
    "iterated_integral": "integral(0 to 1) integral(0 to 1) x*y dy dx",
    "outer_variable": "x",
    "inner_variable": "y",
    "outer_limits": [
      0.0,
      1.0
    ],
    "inner_limits": [
      "0",
      "1"
    ],
    "region_type": "rectangular",
    "value": 0.25,
    "estimated_error": 0.0,
    "average_value": 0.25,
    "method": "composite Simpson with Richardson refinement"
  }
}
```

`result` holds the tool output. Errors come back as
`application/problem+json` with `type`, `title`, `status`, and `detail`.

### Getting a key

If `MINIWEBTOOL_API_KEY` is not already in the environment, stop and ask me for
one — do not sign up, scrape, or guess a key. Free keys: https://api.miniwebtool.com/dashboard/
Save as .cursor/rules/miniwebtool-double-integral-calculator.mdc
---
description: MiniWebtool Double Integral Calculator API — usage and credit budget
globs:
alwaysApply: false
---

# MiniWebtool Double Integral Calculator API

## The API

**Double Integral Calculator** — Evaluate an iterated double integral over rectangular or variable-limit regions.

- Live endpoint: `POST https://api.miniwebtool.com/v1/tools/double-integral-calculator/run` — costs 1 credit
- Dry run: `POST https://api.miniwebtool.com/v1/tools/double-integral-calculator/dry-run` — costs 0 credits, same auth and validation
- Auth: `Authorization: Bearer <MINIWEBTOOL_API_KEY>`
- Content type: `application/json`
- Tool version: `2026-04-22` (output shape is stable within a major version)
- Full machine-readable spec: `https://api.miniwebtool.com/v1/openapi.json`

### Request body

| field | type | required | notes |
|---|---|---|---|
| `function` | str | no | integrand (default `x*y`) |
| `outer_variable` | str | no | single letter, integrated last (default `x`) |
| `inner_variable` | str | no | single letter, integrated first (default `y`) |
| `outer_lower` | str | no | constant lower limit (default `0`) |
| `outer_upper` | str | no | constant upper limit (default `1`) |
| `inner_lower` | str | no | may depend on the outer variable (default `0`) |
| `inner_upper` | str | no | may depend on the outer variable (default `1`) |

Example request body:

```json
{
  "function": "x*y",
  "outer_variable": "x",
  "inner_variable": "y",
  "outer_lower": "0",
  "outer_upper": "1",
  "inner_lower": "0",
  "inner_upper": "1"
}
```

### Response envelope

```json
{
  "request_id": "req_01H…",
  "tool": "double-integral-calculator",
  "tool_version": "2026-04-22",
  "credits_used": 1,
  "result": {
    "integrand": "x*y",
    "iterated_integral": "integral(0 to 1) integral(0 to 1) x*y dy dx",
    "outer_variable": "x",
    "inner_variable": "y",
    "outer_limits": [
      0.0,
      1.0
    ],
    "inner_limits": [
      "0",
      "1"
    ],
    "region_type": "rectangular",
    "value": 0.25,
    "estimated_error": 0.0,
    "average_value": 0.25,
    "method": "composite Simpson with Richardson refinement"
  }
}
```

`result` holds the tool output. Errors come back as
`application/problem+json` with `type`, `title`, `status`, and `detail`.

### Getting a key

If `MINIWEBTOOL_API_KEY` is not already in the environment, stop and ask me for
one — do not sign up, scrape, or guess a key. Free keys: https://api.miniwebtool.com/dashboard/

## Credit budget — follow these exactly

This API is metered. A live call costs 1 credit; the free tier
includes 1,000 credits per month. Burning them during integration is the
single most common way to waste this key.

1. **Do all wiring against `/dry-run`, which costs 0 credits.** It validates the
   payload for real and returns a sample result with the exact response shape.
   Iterate there until your request builds and your parsing works.
2. **Make at most ONE live `/run` call** — a single end-to-end confirmation once
   dry-run passes. Print the result, then stop.
3. **Never call the API from unit tests, examples, or a retry loop.** Assert
   against the sample response captured from `/dry-run` instead.
4. **On 4xx, fix the payload — do not retry.** The error body is RFC 7807
   `application/problem+json` and says exactly what is wrong.
5. **On 429, honour `Retry-After`** and back off; do not tighten the loop.
6. **Read `X-MWT-Credits-Remaining`** on every response. If it drops below 50,
   stop making live calls and tell me.
7. If the integration needs repeated calls at runtime, **cache by input** — this
   tool is deterministic, so the same input always returns the same output.
Skip the HTTP client — connect the agent directly.
Add the MiniWebtool MCP server to this project, then use the
`double-integral-calculator` tool from it.

Server URL: https://api.miniwebtool.com/v1/mcp
Transport: HTTP (Streamable HTTP / JSON-RPC 2.0)
Auth header: `Authorization: Bearer <MINIWEBTOOL_API_KEY>`

Claude Code:

```bash
claude mcp add --transport http miniwebtool https://api.miniwebtool.com/v1/mcp \
  --header "Authorization: Bearer $MINIWEBTOOL_API_KEY"
```

Or add it to the MCP config file your client uses:

```json
{
  "mcpServers": {
    "miniwebtool": {
      "type": "http",
      "url": "https://api.miniwebtool.com/v1/mcp",
      "headers": { "Authorization": "Bearer ${MINIWEBTOOL_API_KEY}" }
    }
  }
}
```

Once connected, `tools/list` exposes `double-integral-calculator` with its JSON Schema, so you can
call it directly instead of writing an HTTP client.

## Credit budget — follow these exactly

This API is metered. A live call costs 1 credit; the free tier
includes 1,000 credits per month. Burning them during integration is the
single most common way to waste this key.

1. **Do all wiring against `/dry-run`, which costs 0 credits.** It validates the
   payload for real and returns a sample result with the exact response shape.
   Iterate there until your request builds and your parsing works.
2. **Make at most ONE live `/run` call** — a single end-to-end confirmation once
   dry-run passes. Print the result, then stop.
3. **Never call the API from unit tests, examples, or a retry loop.** Assert
   against the sample response captured from `/dry-run` instead.
4. **On 4xx, fix the payload — do not retry.** The error body is RFC 7807
   `application/problem+json` and says exactly what is wrong.
5. **On 429, honour `Retry-After`** and back off; do not tighten the loop.
6. **Read `X-MWT-Credits-Remaining`** on every response. If it drops below 50,
   stop making live calls and tell me.
7. If the integration needs repeated calls at runtime, **cache by input** — this
   tool is deterministic, so the same input always returns the same output.
how-to call double-integral-calculator
  1. Get an API key
    Get an API key. Sign up free for a bearer token. 1,000 credits/month, no card.
  2. Assemble the JSON body
    Required fields: . See the Request body table above for the full schema.
  3. POST to https://api.miniwebtool.com/v1/tools/double-integral-calculator/run
    Headers: Authorization: Bearer <key> and Content-Type: application/json. Copy-paste snippets in 7 languages are under Code examples.
  4. Parse the response
    Envelope: {request_id, tool, tool_version, credits_used, result}. The typed output lives in result.
faq frequently asked
What does the Double Integral Calculator API do?+

Evaluate an iterated double integral over rectangular or variable-limit regions. Use it to add this utility without rebuilding formulas, validation, examples, and documentation.

How much does one call to Double Integral Calculator cost?+

1 credit per successful call. Failed calls (validation errors, 5xx) don't bill.

What parameters does the Double Integral Calculator API require?+

Required fields: (none). Full schema at /v1/openapi.json.

Is the Double Integral Calculator API deterministic?+

Yes — same input, same output, forever. Tool version 2026-04-22; output shape is stable within a major version.

Can an AI agent call the Double Integral Calculator API?+

Yes. Connect to https://api.miniwebtool.com/v1/mcp over the Model Context Protocol. The agent discovers `double-integral-calculator` with its JSON Schema automatically, so the model can plan while MiniWebtool handles the exact tool result.

What format does the API return?+

JSON with a stable envelope: `{request_id, tool, tool_version, credits_used, result}`. Errors are RFC 7807 `application/problem+json`.

mcp use from an AI agent

Every endpoint on this site is also exposed via the Model Context Protocol at https://api.miniwebtool.com/v1/mcp. Claude, Cursor, and any MCP-capable agent can discover this tool (double-integral-calculator) and its JSON Schema automatically — no client codegen. 

# List tools the agent can call
curl -s https://api.miniwebtool.com/v1/mcp \
  -H 'Content-Type: application/json' \
  -d '{"jsonrpc":"2.0","id":1,"method":"tools/list"}' \
 | jq '.result.tools[] | select(.name=="double-integral-calculator")'
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