Double Integral Calculator (Rectangular Region)
Calculus, Multivariable
Intro: Evaluates double integrals over simple rectangular regions by performing iterated integration with respect to one variable at a time.
Worked example
- Evaluate $∬_R x y dA$ where $R = [0,2]×[1,3]$.
- We want $\iint_R x y\,dA$ over the rectangle $0 \le x \le 2$, $1 \le y \le 3$.
- Choose the order dy dx: $\iint_R x y\,dA = \int_{x=0}^{2} \int_{y=1}^{3} x y\,dy\,dx$.
- Treat x as a constant in the inner integral and integrate with respect to y: $\int_{1}^{3} x y\,dy = x\int_1^3 y\,dy$.
- Compute the inner integral: $\int_1^3 y\,dy = \left.\dfrac{y^2}{2}\right|_1^3 = (9/2)-(1/2)=8/2=4$.
- So the inner result is $x\cdot4=4x$. Now we integrate $4x$ with respect to x from 0 to 2: $\int_0^2 4x\,dx$.
- Compute: $\int_0^2 4x\,dx = 4 \cdot \left.\dfrac{x^2}{2}\right|_0^2 = 4\cdot(4/2 - 0) = 4\cdot2 = 8$.
- Thus $\iint_R x y\,dA = 8$.
- Answer: $\boxed{8}$.
FAQs
Can it handle variable bounds?
This version focuses on rectangular regions with constant bounds. More general regions can be added as an advanced feature.
Why choose MathGPT?
- Get clear, step-by-step solutions that explain the “why,” not just the answer.
- See the rules used at each step (power, product, quotient, chain, and more).
- Optional animated walk-throughs to make tricky ideas click faster.
- Clean LaTeX rendering for notes, homework, and study guides.
How this calculator works
- Type or paste your function (LaTeX like
\sin,\lnworks too). - Press Generate a practice question button to generate the derivative and the full reasoning.
- Review each step to understand which rule was applied and why.
- Practice with similar problems to lock in the method.