Question

​​​​​​​
The area of the region enclosed by the curve of x=49-y² and the line x=32 is

Answer 1

The area of the region enclosed by the curve x = 49 - y² and the line x = 32 is 34√17 square units.

Step-by-step explanation:

To find the area of the region enclosed by the curve x = 49 - y² and the line x = 32, we first graph these two equations to see the region. The curve x = 49 - y² is a downward-opening parabola that opens to the right and the line x = 32 is a vertical line intersecting the curve at x = 32. The intersection points are found by setting the two equations equal to each other and solving for y:

49 - y² = 32

y² = 17

y = ±√17

Using these intersection points, we can set up the integral to find the area:

∫(from √17 to -√17) (49 - y² - 32) dy

Simplifying and integrating, we get:

∫(from √17 to -√17) (17 - y²) dy

This integral evaluates to 34√17 square units, which is the area of the region enclosed by the curve and the line.