Area Between Two Curves:?

Question

Find the area of the shaded region bounded by y = 7 x and y = x (sqrt{(15^{2}-x^{2})}. )

A=

Answer 1

y = 7x and y = x√(15² - x²)

7x - x√(15² - x²) = 0

x[7 - √(15² - x²)] = 0

.x = 0 or [7 - √(15² - x²)] = 0

Take [7 - √(15² - x²)] = 0

Subtract -7 from each side and multiply each side by negative one.

√(15² - x²) = 7

Take square each side.

225 - x² = 49 ⇒ x² = 225-49=176⇒x =√176 ⇒ x = ±13.26

The two equation interval is (13.26, 0, -13.26)

In interval 0<x<13.26, y₂>y₁ which makes enclosed area A₁.

In interval -13.26<x<0,y₁>y₂ which makes enclosed area A₂.

The area between two curves is A₁= ʃ(y₂-y₁)dx

y₂ - y₁ =

Integrals with roots

y₁ - y₂ =

Total area is A = A₁ + A₂ = 394.666+394.666 = 789.332 unit².