Question

2. Perform the following tasks: (a) Evaluate the integral: $\iint_{\mathbb{R}^2} e^{-x^2 - y^2} dA$ (b) Use your answer to part (a) to evaluate the integral: $\int_{-\infty}^{\infty} e^{-x^2} dx$ (Hint: $\int_a^b \int_c^d f(x)g(y) dx dy = \int_a^b g(y) dy \int_c^d f(x) dx)$

          2. Perform the following tasks:
(a) Evaluate the integral:
$\iint_{\mathbb{R}^2} e^{-x^2 - y^2} dA$
(b) Use your answer to part (a) to evaluate the integral:
$\int_{-\infty}^{\infty} e^{-x^2} dx$
(Hint: $\int_a^b \int_c^d f(x)g(y) dx dy = \int_a^b g(y) dy \int_c^d f(x) dx)$

2. Perform the following tasks:
(a) Evaluate the integral:
∬ℝ^2 e^-x^2 - y^2 dA
(b) Use your answer to part (a) to evaluate the integral:
∫-∞^∞ e^-x^2 dx
(Hint: ^b ^d f(x)g(y) dx dy = ^b g(y) dy ^d f(x) dx)

Added by David L.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Shu-Ting Huang

Step 1

The integral - y dA represents the integral of the function -y over some region in the xy-plane. The differential area element dA represents an infinitesimally small area in the xy-plane. To determine the limits of integration and the differential area element, Show more…

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2. Perform the following tasks: (a) Evaluate the integral: - y dA (b) Use your answer to part (a) to evaluate the integral: (Hint: f(x)g(y) dxdy f(x) dx