Application of Integration: Plane Area

Question

Show your mathematical maturity by giving COMPLETE complete solution to each of the

following items.

1. Determine the area bounded by the curves y = 6x − x² and y = x² − 2x.

2. Determine the tangent to the curve 3y² = x³at (3,3) and calculate the area of the triangle bounded by the tangent line, the x-axis and the line x = 3.

3. Find the area of the curve r² = a²cos2θ.

Answer 1

1.

(6x - x²) - (x² - 2x) = 8x - 2x² = 2x(4 - x)

2x(4 - x) = 0 at 0, 4

I[0, 4] is integrate from 0 to 4, E[0, 4] is evaluate at 0 and 4.

I[0, 4] 8x - 2x² dx = 4x² - 2/3x³ E[0,4] = 4*4² - 2/3*4³ = 64 - 128/3 = 64/3

2.

3y² = x³

6ydy = 3x²dx

dy/dx = 3x²/6y = 3*32/(6*3) = 27/18 = 3/2

Thus, the tangent line is

3 = 3/2 * 3 + b

3 - 9/2 = b

b = -3/2

We have a triangle consisting of

Thus, we have y = 3/2x - 3/2, y = 0, and x = 3

3/2x - 3/2 = 0

3/2x = 3/2

x = 1

Thus, we have a triangle with points (1, 0), (3, 0), and (3, 3)

This has area 1/2 bh = 1/2(3-1)(3-0) = 3

3.

r² = a² cos2θ

r² >= 0 and a² >= 0

Thus, cos2θ >= 0

cosθ >= 0 on [-π/2, π/2]

Thus, cos2θ >= 0 on [-π/4, π/4] and [3π/4, 5π/4]

r² = a² cos2θ for r = a √cos2θ

Then, letting r dr dθ = dydx, we have

I[-π/4, π/4] I[0, a √cos2θ] r dr dθ + I [3π/4, 5π/4] I[0, a √cos2θ] r dr dθ =

I[-π/4, π/4] 1/2 r² E[0, a √cos2θ] dθ + I[3π/4, 5π/4] 1/2 r² E[0, a √cos2θ] dθ =

I[-π/4, π/4] 1/2 a² cos2θ dθ + I[3π/4, 5π/4] 1/2 a² cos2θ dθ =

1/4 a² sin2θ E[-π/4, π/4] + 1/4 a² sin2θ E[3π/4, 5π/4] =

1/4 a² (1 - -1) + 1/4 a² (1 - -1) =

a²