Question

Evaluate the double integral. 7x cos(y) da, d is bounded by y = 0, y = x2, x = 7 d

Answer 1

The double integral. 7x cos(y) da, d is bounded by y = 0, y = x2, x = 7 d is given as

$\int _D 7xcosydA =7/2(-cos49+1)$

What is the double integral 7x cos(y) da, d is bounded by y = 0, y = x2, x = 7 d?

Generally, the equation for is  mathematically given as

The area denoted by the letter D that is bordered by y=0, y=x2, and x=7

The equation for the X-axis is y=0.

y=x² ---> (y-0) = (x-0)²

Therefore, the equation of a parabola is y = x2, and the vertex of the parabola is located at the point (0,0), and the axis of the parabola is parallel to the Y axis.

The equation for a straight line that is parallel to the Y-axis and passes through the point (7,0) is x=7.

$\int _D 7xcosydA\\\\\int^7_0 \int^x^2 _0 7xcosydA$

Integrating we have

$7/2 \int^7_0 (2xsinx^2)dx$

If x equals zero, then we know that u equals zero as well.

When x equals seven, we know that u=72=49.

Therefore, by changing x2=u into our integral, it becomes from

 $7/2 \int^7_0 (2xsinx^2)dx$

$7/2 \int^49_0 sin u dx$

Hence

=7/2(-cos49+1)

In conclusion,

$\int _D 7xcosydA =7/2(-cos49+1)$

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