The following integrals calculate areas of regions in the xy-plane. Say what shape each integral gives the area of. Be specific; for example, if the shape is a rectangle, give the base and height of the rectangle. Include a sketch of the region, showing the variable of integration, an arbitrary slice, and any other relevant quantities. a. ∫ 3xdx
b. ∫ √15-y^2dy
c. ∫ √81- x^2 dx
d. ∫ (5- y/7)dy

a.

b.

c.

d.

Step-by-step explanation:

P.S - The exact question is -

a.

y = 3x

Shape of the given integral = A Triangle

Area of Triangle = = ×1×3 =

The graph of integral is as follows :

b.

x =

⇒x² = 15 - y²

⇒x² + y² = 15

⇒x² + y² = (√15)²

Shape of the integral = Quarter circle of radius √15

Area of Quarter circle = = ××r² = ××(√15)² =

The graph of the following integral is as follows :

c.

y =

⇒y² = 81 - x²

⇒y² + x² = 81

⇒y² + x² = 9²

Shape of the integral = Semi circle of radius 9

Area of semi circle = = ××r² =  ××9² =

The graph of the following integral is as follows :

d.

x = 5 -

⇒ = 5 - x

⇒y = 35 - 7x

At y = 0 , x = 5

At y = 7 , x = 4

Shape of the integral = A triangle

Area of Triangle = ×(5-4)×7 =  ×(1)×7 =

The graph of the integral is as follows :