Double integral (dblquad) in SciPy - Step-by-Step Execution

Concept Flow - Double integral (dblquad)

Define function f(y,x)

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Set integration limits for y

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Set integration limits for x

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Call dblquad(f, x_lower, x_upper, y_lower(x), y_upper(x))

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Compute inner integral over y

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Compute outer integral over x

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Return total integral and error estimate

The process defines the function and integration limits, then computes the inner integral over y for each x, followed by the outer integral over x, returning the total integral value.

Execution Sample

SciPy

from scipy.integrate import dblquad

f = lambda y, x: x * y
result, error = dblquad(f, 0, 2, lambda x: 0, lambda x: 3)
print(result)

This code calculates the double integral of f(y,x) = x*y over x from 0 to 2 and y from 0 to 3.

Execution Table

Step	Action	x value	Inner integral over y	Partial result	Notes
1	Start dblquad	-	-	-	Begin integration over x from 0 to 2
2	Evaluate inner integral at x=0	0	Integral of 0*y dy from 0 to 3 = 0	0	f(y,0) = 0 for all y
3	Evaluate inner integral at x=1	1	Integral of 1*y dy from 0 to 3 = 4.5	Partial outer integral accumulates	Integral y dy = y^2/2 from 0 to 3 = 4.5
4	Evaluate inner integral at x=2	2	Integral of 2*y dy from 0 to 3 = 9	Partial outer integral accumulates	Double the previous inner integral value
5	Compute outer integral over x	-	-	Total integral = 9	Integral of inner results over x from 0 to 2
6	Return result and error	-	-	9.0	Integration complete

💡 Integration completes after evaluating inner integrals at sample points and summing over x range

Variable Tracker

Variable	Start	After Step 2	After Step 3	After Step 4	Final
x	-	0	1	2	-
inner_integral	-	0	4.5	9	-
result	-	-	-	-	9.0

Key Moments - 3 Insights

Why does the function f take arguments in the order (y, x) instead of (x, y)?

In dblquad, the function must be defined as f(y, x) because the integration is done first over y (inner integral), then over x (outer integral). This matches the calling convention shown in the execution_table steps.

How are the limits for y defined as functions of x?

The limits for y are given as functions of x (here constant 0 and 3). This allows flexible integration regions. The execution_table shows these limits used at each x step.

Why do we see partial inner integrals before the final result?

dblquad computes the inner integral over y for each x, then sums these results over x. The execution_table rows 2-4 show these partial inner integrals accumulating to the final total.

Visual Quiz - 3 Questions

Look at the execution_table, what is the inner integral value at x=1?

A 0

B 9

C 4.5

D 3

💡 Hint

Check row 3 in the execution_table under 'Inner integral over y'

At which step does the outer integral complete?

A Step 4

B Step 5

C Step 6

D Step 3

💡 Hint

Look for the step where the total integral is computed over x in the execution_table

If the upper limit for y changed from 3 to 4, how would the inner integral at x=1 change?

A It would become 8

B It would stay 4.5

C It would become 6

D It would become 9

💡 Hint

Recall inner integral at x=1 is integral of y dy from 0 to upper limit; use formula y^2/2

Concept Snapshot

dblquad syntax: dblquad(f, x_lower, x_upper, y_lower(x), y_upper(x))
Function f(y,x) integrates first over y, then x.
Limits for y can depend on x.
Returns integral value and error estimate.
Used for calculating area/volume under surfaces.

Full Transcript

This visual execution traces how the scipy dblquad function calculates a double integral. We start by defining the function f(y,x) and the integration limits for x and y. The integration proceeds by first computing the inner integral over y for fixed x values, then summing these results over x. The execution table shows step-by-step inner integral values at sample x points and the accumulation to the final integral result. Key points include the argument order of f(y,x), the use of limits for y as functions of x, and the two-stage integration process. The visual quiz tests understanding of these steps and how changing limits affects results.