Question

Evaluate iint R (5 - 2x - y) * dA , where R is given by. y = 0 , x + 2y = 3 , x = y ^ 2

Answer 1

We need to evaluate the following integral.

\(I ={\int \int_A } (5 - 2x - y) dx\ dy\)

where A is given by y = 0, x + 2y = 3 and x = y²

First of all plot the graphs for given equations and find the limits for the integral as the region A is given by provided equations.

Interpreting region A, we get the following limits:

\(I = \int \limits_0^1 \int \limits_{y^2}^{3- 2y} (5 - 2x - y) dx\ dy\)

Solving the inner integral, w.r.t. x keeping y's as constant.

Solving the integral w.r.t. y

So the required answer is,

\(I ={\int \int_A } (5 - 2x - y) dx\ dy = \frac {217} {60}\)