Spring 2006 - Nagy's Class - Exam 1

Spring 2006 - Nagy's Class - Exam 1 - Name Section Number...

Info icon This preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Name: Section Number: TA Name: Section Time: Math 20B. Midterm Exam 1 April 28, 2006 No calculators or any other devices are allowed on this exam. Write your solutions clearly and legibly; no credit will be given for illegible solutions. Read each question carefully. If any question is not clear, ask for clarification. Answer each question completely, and show all of your work. 1. (10 points) Evaluate the following integrals. π/3 (a) 0 (b) sin(θ) dθ cos2 (θ) 1+x dx 1 + x2 (a) Substitute u = cos(θ), so du = − sin(θ) dθ, and u(0) = 1, u(π/3) = 1/2. Then π/3 0 sin(θ) dθ = cos2 (θ) 1/2 1 − du = u2 1 u−2 du = 1/2 1 (−1) u−1 1 1/2 = −(1 − 2) = 1. (b) 1+x dx = 1 + x2 1 dx + 1 + x2 x dx 1 + x2 Integrate the first term directly, and in the second term do the substitution u = 1 + x 2 , with du = 2x dx, then 1+x dx = arctan(x) + 1 + x2 1 du , 2 u 1 ln(u) + c, 2 = arctan(x) + ln( 1 + x2 ) + c. = arctan(x) + # 1 2 3 4 Σ Score 2. (10 points) Find the area of the shaded region y sin(x) 1 cos(x) pi/2 x Split the integration region in two intervals, [0, π/4] and [π/4, π/2]. Then, the area A of the shaded region is π/4 A = π/2 sin(x) dx + 0 cos(x) dx, π/4 π/4 π/2 − cos(x)|0 + sin(x)|π/4 , √ √ 2 2 −1 + 1− , = − 2 2 √ = 2 − 2. = 0 V = h Then, ab 2 ab 1 y dy = 2 h3 2 h h 3 A(y) = Then x(y) = ⇒ V = 1 abh. 3 ab 2 y . h2 (a/2) y. h Analogously, the x(y) function is a line given by z(y) = (b/2) y. h Concentrate in the x = 0 plane, then the top of the pyramid is a line in the zy-plane, passing through the origin, so the function z(y) is given by This area is given by A(y) = 2z(y) 2x(y). ¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦§ § § § § § § § § § § § § § § § § § § § § § § § § § § § § § § ¦§¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤ ¦§¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¦¦ § § § § § § § § § § § § § § § § § § § § § § § § § § § § § § § ¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦§ ¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤ §§¤§¦¤§¦¤§¦¤§¦¤§¦¤§¦¤§¦¤§¦¤§¦¤§¦¤§¦¤§¦¤§¦¤§¦¤§¦¤§¦¤§¦¤§¦¤§¦¤§¦¤§¦¤§¦¤§¦¤§¦¤§¦¤§¦¤§¦¤§¦¤§¦¤§¦¤¦¤ ¦¦¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤§¤¦¦ §¦§ §¦§ §¦§ §¦§ §¦§ §¦§ §¦§ §¦§ §¦§ §¦§ §¦§ §¦§ §¦§ §¦§ §¦§ §¦§ §¦§ §¦§ §¦§ §¦§ §¦§ §¦§ §¦§ §¦§ §¦§ §¦§ §¦§ §¦§ §¦§ §¦§ §¦§ ¦¦§§¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¦¦ ¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤ § § § § § § § § § § § § § § § § § § § § § § § § § § § § § § § ¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§§ ¦¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦§¤ ¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¦¦¦ ¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦§ § § § § § § § § § § § § § § § § § § § § § § § § § § § § § § ¦§§¤§¦¤§¦¤§¦¤§¦¤§¦¤§¦¤§¦¤§¦¤§¦¤§¦¤§¦¤§¦¤§¦¤§¦¤§¦¤§¦¤§¦¤§¦¤§¦¤§¦¤§¦¤§¦¤§¦¤§¦¤§¦¤§¦¤§¦¤§¦¤§¦¤§¦¤¦¤ ¦¦§§¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¦¦ ¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§§ § § § § § § § § § § § § § § § § § § § § § § § § § § § § § § ¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤ ¦¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§§¤ ¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¦¦ § §§¦ §§¦ §§¦ §§¦ §§¦ §§¦ §§¦ §§¦ §§¦ §§¦ §§¦ §§¦ §§¦ §§¦ §§¦ §§¦ §§¦ §§¦ §§¦ §§¦ §§¦ §§¦ §§¦ §§¦ §§¦ §§¦ §§¦ §§¦ §§¦ §§¦ §¦ ¦¦§§¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¦¦ ¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤ ¦§ ¦§ ¦§ ¦§ ¦§ ¦§ ¦§ ¦§ ¦§ ¦§ ¦§ ¦§ ¦§ ¦§ ¦§ ¦§ ¦§ ¦§ ¦§ ¦§ ¦§ ¦§ ¦§ ¦§ ¦§ ¦§ ¦§ ¦§ ¦§ ¦§ ¦§ ¦§¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¦¦¦ ¦¦§§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤ ¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤ ¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦§ § § § § § § § § § § § § § § § § § § § § § § § § § § § § § § ¦¦§§¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¦¦ ¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§¤¦§§ § § § § § § § § § § § § § § § § § § § § § § § § § § § § § § ¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦¤ ££¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¥¦§¤ ¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤£¦¦ ¥¦§¤¥¥¦§¤¥¥¦§¤¥¥¦§¤¥¥¦§¤¥¥¦§¤¥¥¦§¤¥¥¦§¤¥¥¦§¤¥¥¦§¤¥¥¦§¤¥¥¦§¤¥¥¦§¤¥¥¦§¤¥¥¦§¤¥¥¦§¤¥¥¦§¤¥¥¦§¤¥¥¦§¤¥¥¦§¤¥¥¦§¤¥¥¦§¤¥¥¦§¤¥¥¦§¤¥¥¦§¤¥¥¦§¤¥¥¦§¤¥¥¦§¤¥¥¦§¤¥¥¦§¤¥¥¦§¤¥£ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ ££¥¥¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤££ ¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤ £¥ £¥ £¥ £¥ £¥ £¥ £¥ £¥ £¥ £¥ £¥ £¥ £¥ £¥ £¥ £¥ £¥ £¥ £¥ £¥ £¥ £¥ £¥ £¥ £¥ £¥ £¥ £¥ £¥ £¥ £¥ £¥¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤££ £¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¥ ¤££¥¤££¥¤££¥¤££¥¤££¥¤££¥¤££¥¤££¥¤££¥¤££¥¤££¥¤££¥¤££¥¤££¥¤££¥¤££¥¤££¥¤££¥¤££¥¤££¥¤££¥¤££¥¤££¥¤££¥¤££¥¤££¥¤££¥¤££¥¤££¥¤££¥¤££¥¤ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ££¥¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤£££ ¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤ £¥¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤ £¥¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤££ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ £¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¥ ¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤ ¥¥¤¥£¤¥£¤¥£¤¥£¤¥£¤¥£¤¥£¤¥£¤¥£¤¥£¤¥£¤¥£¤¥£¤¥£¤¥£¤¥£¤¥£¤¥£¤¥£¤¥£¤¥£¤¥£¤¥£¤¥£¤¥£¤¥£¤¥£¤¥£¤¥£¤¥£¤£¤ ££¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¥¤££ ¤¥£¤¥£¤¥£¤¥£¤¥£¤¥£¤¥£¤¥£¤¥£¤¥£¤¥£¤¥£¤¥£¤¥£¤¥£¤¥£¤¥£¤¥£¤¥£¤¥£¤¥£¤¥£¤¥£¤¥£¤¥£¤¥£¤¥£¤¥£¤¥£¤¥£¤¥£¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ££¥¥¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤££ ¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤ £¥ £¥ £¥ £¥ £¥ £¥ £¥ £¥ £¥ £¥ £¥ £¥ £¥ £¥ £¥ £¥ £¥ £¥ £¥ £¥ £¥ £¥ £¥ £¥ £¥ £¥ £¥ £¥ £¥ £¥ £¥ ££¥¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤ ¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤£££ ¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ £¥¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤ ££¥¥¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤££ ¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤ £¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤£¤ ¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤£££ ¨ ¥ ¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¨© © ¥© ¥© ¥© ¥© ¥© ¥© ¥© ¥© ¥© ¥© ¥© ¥© ¥© ¥© ¥© ¥© ¥© ¥© ¥© ¥© ¥© ¥© ¥© ¥© ¥© ¥© ¥© ¥© ¥© ¥© £¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¤£¥¥ ¤ ¨¤¨¤¨¤¨¤¨¤¨¤¨¤¨¤¨¤¨¤¨¤¨¤¨¤¨¤¨¤¨¤¨¤¨¤¨¤¨¤¨¤¨¤¨¤¨¤¨¤¨¤¨¤¨¤¨¤¨¤¨¤ a ¢ ¢ ¢ ¢ ¢¢¢¡¡¡¡¡ ¢¢ ¡ ¡ ¡ ¡ ¡ ¡ ¢¡ ¢¡ ¢¡ ¢¡ ¢ ¢ ¢ ¢ ¢¡¢¡¢¡¢¡¢¡ ¢ ¢ ¢ ¡¢¡¢¡¢¡¢¡ ¡ ¡ ¡ ¡ ¡ ¡ ¢¡ ¢¡ ¢¡ ¢¡ ¢¢¢¡¡¡¡¡ ¢¢ ¢ ¡ ¡ ¡ ¡ ¡ ¡ ¢¡ ¢¡ ¢¡ ¢¡ ¢ ¢ ¢ ¢ ¢¡¢¡¢¡¢¡¢¡ ¢ ¢ ¢ ¡¢¡¢¡¢¡¢¡ ¡ ¡ ¡ ¡ ¡ ¡ ¢¡ ¢¡ ¢¡ ¢¡ ¢¢¢¡¡¡¡¡ ¢¢ ¢ ¡ ¡ ¡ ¡ ¡ ¡ ¢¡ ¢¡ ¢¡ ¢¡ ¢ ¢ ¢ ¢ ¢¡¢¡¢¡¢¡¢¡ ¢ ¢ ¢ ¡¢¡¢¡¢¡¢¡ ¡ ¡ ¡ ¡ ¡ ¡ ¢¡ ¢¡ ¢¡ ¢¡ ¢¢¢¡¡¡¡¡ ¢¢ ¢ ¡ ¡ ¡ ¡ ¡ ¡ ¢¡ ¢¡ ¢¡ ¢¡ ¢ ¢ ¢ ¢ ¢¡¢¡¢¡¢¡¢¡ ¢ ¢ ¢ ¡¢¡¢¡¢¡¢¡ ¡ ¡ ¡ ¡ ¡ ¡ ¢¡ ¢¡ ¢¡ ¢¡ ¢¢¢¡¡¡¡¡ ¢¢ ¢ ¡ ¢¡ ¢¡ ¢¡ ¢¡ ¢ ¢ ¢ ¢ ¡ ¡ ¡ ¡ ¡ ¡¡¡¡¡ ¢¡¢¡¢¡¢¡¢¡ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¡ ¡ ¡ ¡ ¡ ¡ ¢¡ ¢¡ ¢¡ ¢¡ ¢¢¢¡¡¡¡¡ ¢¢ ¢ ¡ ¢¡ ¢¡ ¢¡ ¢¡ ¢ ¢ ¢ ¢ ¡ ¡ ¡ ¡ ¡ ¡¡¡¡¡ ¢¡¡¡¡¡¢ ¢ ¡¢ ¡¢ ¡¢ ¡¢ ¡ ¢ ¢ ¢ ¢ ¢ ¢¡ ¢¡ ¢¡ ¢¡ ¢¡ ¡¡¡¡¡ ¢ ¢ x y y h b b/2 z h y b z The idea is to find the area of rectangular regions perpendicular to the y-axis, as function of y. a x y h b z 3. (10 points) Find the volume of a pyramid with rectangular base of sides a and b, and height h. 4. (8 points) Compute both (1 + i)8 and (1 + i)10 . z = 1+i ⇒ √ √ r = 1 + 1 = 2, tan(θ) = 1 ⇒ θ = π/4. z = 21/2 [cos(π/4) + i sin(π/4)] ⇒ z 8 = 28/2 [cos(8π/4) + i sin(8π/4)] = 24 [cos(2π) + i sin(2π)] = 16. (1 + i)8 = 16. z = 21/2 [cos(π/4) + i sin(π/4)] ⇒ ⇒ ⇒ z 10 = 210/2 [cos(10π/4) + i sin(10π/4)] = 25 [cos(5π/2) + i sin(5π/2)] ⇒ z 10 = 32[cos(π/2) + i sin(π/2)] = 32 i. (1 + i)10 = 32 i. ⇒ ...
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern