Let R be the region in the plane bounded by the lines y = x^2, x = 0, and y = 4. Evaluate the double integral xe^(y^2) dA

Question

for the limits of integration I got x^2 to 4 and 0 to 2.

Answer 1

               y=4

(0,4) --------------- (2,4)

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x=0  |              / y=x²

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        --

      (0,0)

 

We have to integrate out x first, since the indefinite integral ∫ exp(y²) dy has no closed form.

 

Therefore we must use horizontal slices. At level any level 0 ≤ y ≤ 4, we have 0 ≤ x ≤ √y

 

Thus the iterated integral is:

 

∫₀⁴ ∫₀^√y x exp(y²) dx dy

 

= ∫₀⁴ x² exp(y²)/2 |₀^√y dy

 

= (1/4) ∫₀⁴ 2y exp(y²) dy

 

= (1/4) ∫₀¹⁶ e^u du    ← u-substitution: u=y², du = 2y dy

 

= e^u/4 |₀¹⁶ = (e¹⁶ - 1)/4