Find the area of the region bounded by following

Question

Sketch the region enclosed by the given curves and decide whether to integrate with respect to x or y. Find the area of the region bounded by:

2y=5sqrtx

y=3

2y+4x=9

Answer 1

Video does not show how to find points of intersection. It also does not perform the calculation of the definite integral. The video simply suggests that integrating with respect to y will be easier.

Answer 2

Plot the equations using something like DESMOS:

https://www.desmos.com/calculator

The 2nd and 3rd equations are linear, so I think the integration will be determined by the 1st equation.

Square both sides of the 1st equation to get:

4y2 = 25x

x = (4/25)*y²

This is a parabola opening to the right, with the vertex at (0,0)

Rearrange the 3rd equation to get:

4x = 9 - 2y

x = 9/4 - y/2

You will see that the line y = 3 is outside of the area bounded by the other 2 equations. Therefore, just use those 2 to find the enclosed area. First, solve the pair to get the 2 points of intersection and then use those points for the integration limits.

Combining the equations, you will be integrating this:

x = (4/25)*y² -(9/4 - y² )

x = (29/25)*y² - 9/4