Question

Evaluate the double integral. ∬ D e⁻ʸ² dA,D={(x,y)∣0⩽y⩽3,0⩽x⩽y}

Answer 1

To evaluate the given double integral, we need to express the region D and the integrand in terms of one variable, and then evaluate the resulting single integral. The value of the double integral is (1/2) - (1/2)e^-9.

Step-by-step explanation:

To evaluate the given double integral, we need to express the region D and the integrand in terms of one variable and then evaluate the resulting single integral.

First, let's express the region D:

D = (x,y)

Since x is bounded by 0 and y, we can rewrite the integral as:

∬D e-y² dA = ∫03 ∫0y e-y² dx dy

Now, let's evaluate the integral:

∫0y e-y² dx = xe-y² |0y = ye-y² - 0 = ye-y²

∫03 ye-y² dy = -(1/2)e-y² |03 = -(1/2)e-9 + (1/2)e0 = (1/2) - (1/2)e-9

Therefore, the value of the double integral is (1/2) - (1/2)e-9.