MIT 18.01 Single Variable Calculus Exam 4, Study notes of Mathematics

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18.01 Single Variable Calculus

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Fall 2006

18.01 Exam 4

Problem 1. (15 points) Evaluate (^) ( 1) 2

dx

∫ x x +

Problem 2. (15 points) Evaluate ∫(ln x ) x dx^2

Problem 3. (20 points) Use a trigonometric substitution to evaluate (Be careful evaluating the limits)

1 (^0) (4 3 )3/ 3

dx

∫ + x

Problem 4. a. (10 points) Find an integral formula for the arc length of the curve

y = 2 x + 1 for 0≤ x ≤ 1. Do not evaluate.

b. (10 points) Find an integral formula for the surface area of the curve in part (a) rotated around the x -axis. Simplify the integrand and evaluate the integral.

Problem 5. a. (7 points) Sketch the spiral r = θ^21 0 ≤ θ ≤ 3 ∏. Say how many times the curve meets the x-axis counting θ = 0 as the first times, and mark those points with X-s. (Your sketch need not be accurate to scale.)

b. (8 points) On your picture, shade in the region 0 ≤ r ≤ θ^2 , 0 ≤ θ ≤ 2 ∏, and find its area.

Problem 6. a. (10 points) Find the equation in polar coordinates for the line

y = x – 1 in the form r= ∫( ) θ

b. (5 points) Find the range of θ for the portion of line y=x-1 in the range (^0) ≤ x ≤ ∞. (It helps to draw a picture.)