104 - + 6 sin + + 1 2 sin 2 5 / 4 / 4 + 9 2-6 cos + -1 2...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
MATH 153 Selected Solutions for § 10.4 Exercise 8 : Find the area of the shaded region. - 0.5 0.0 0.5 - 0.5 0.0 0.5 Figure 1. r = sin 2 θ . Solution : We first find the θ values where the curve hits the origin by solving for r = 0. But r = 0 if sin 2 θ = 0, or if 2 θ = 0 . Thus, the values we’re going to use to integrate are 0 and π/ 2. A = Z π/ 2 0 1 2 r 2 = 1 2 Z π/ 2 0 sin 2 2 θ dθ = 1 2 Z π/ 2 0 1 - cos 4 θ 2 = θ 4 - sin 4 θ 16 ± ± ± ± π/ 2 0 = π/ 2 4 - sin 2 π 16 - ( 0 4 - sin 0 16 ) = π 8 . 1
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
2 Exercise 32 : Find the area of the region that lies inside both curves. r = 3 + 2 cos θ, r = 3 + 2 sin θ - 4 - 2 0 2 4 - 4 - 2 0 2 4 Figure 2. The two curves with desired area shaded. Solution : Setting the two curves equal to each other, we see that they in- tersect when sin θ = cos θ , or at θ = π/ 4 , 5 π/ 4. Thus, at these θ values, we switch the function that governs the boundary of the area, and we thus break up the integral there. Notice we integrate from π/ 4 to 9 π/ 4 for simplicity. A = Z 5 π/ 4 π/ 4 1 2 (3 + 2 cos θ ) 2 + Z 9 π/ 4 5 π/ 4 1 2 (3 + 2 sin θ ) 2 = Z 5 π/ 4 π/ 4 9 2 + 6 cos θ + 2 cos 2 θ dθ + Z 9 π/ 4 5 π/ 4 9 2 + 6 sin θ + 2 sin 2 θ dθ = Z 5 π/ 4 π/ 4 9 2 + 6 cos θ + 1 + cos 2 θ dθ + Z 9 π/ 4 5 π/ 4 9 2 + 6 sin θ + 1 - cos 2 θ dθ = ± 9 θ 2
Background image of page 2
Background image of page 3
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: + 6 sin + + 1 2 sin 2 5 / 4 / 4 + 9 2-6 cos + -1 2 sin 2 9 / 4 5 / 4 = 45 8-3 2 + 5 4 + 1 2 - 9 8 + 3 2 + 4 + 1 2 + 81 8-3 2 + 9 4-1 2 - 45 8 + 3 2 + 5 4-1 2 = 11 -12 2 3 Exercise 46 : Find the exact length of the polar curve. r = e 2 , 2 20000 40000 60000 80000 100000 120000-50000-40000-30000-20000-10000 Figure 3. The curve r = with 0 2 . Solution : Note that in the picture above, the function grows so quickly that Mathematica refused to draw it all the way to 2 . This is going to be a large number. L = Z 2 s r 2 + dr d 2 d = Z 2 p e 4 + 4 e 4 d = Z 2 5 e 2 d = 5 2 e 2 2 = 5 2 e 4 - 5 2 320 , 597 ....
View Full Document

This note was uploaded on 07/26/2011 for the course MATH 153 taught by Professor Rempe during the Spring '08 term at Ohio State.

Page1 / 3

104 - + 6 sin + + 1 2 sin 2 5 / 4 / 4 + 9 2-6 cos + -1 2...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online