Worksheet 13 PDF

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Calculus I Fall 2020 - Worksheet 13 Do not use a calculator!

Topics: substitution method, area between curves, net change You must show all work in order to receive credit. A correct answer without work will not receive points. 1. Use the substitution method to solve the following integrals: R (a) 2 sin3 x cos5 x dx R4 (b) 3 2x8x+6 2 +3x dx R (c) x cos(x2 + 1) + x2x+1 dx R2 (d) 1 x2 + e1−x dx 2. Determine the area of the shaded region in the following figures.

(a) Figure 1

(b) Figure 2

3. Determine the area of the region enclosed by y = sin x, y = cos x, x =

π 2,

and the y-axis.

√ 4. A 2,000 liter tank is empty when water starts flowing into it at a rate given by Q′ (t) = 3 t liters per minute. (a) What does Q′ (1) = 3 mean in the context of this problem? (Write a sentence) (b) How much water flows into the tank in the first hour? (c) Find the function Q(t) that gives the amount of water (in liters) in the tank at any time t ≥ 0.

(d) When will the tank be full?

5. The engine on a boat starts at t = 0 and consumes fuel at a rate of c(t) = 5 − e−t liters per hour. How much fuel does it consume in the first two hours? 6. Consider an object moving along a line with a given velocity v. Assume time t is measured in seconds and velocity has unit of m/s. v(t) = 3t2 − 18t + 24 on [0, 5] 1

(a) Determine when the motion is in the positive direction and when it is in the negative direction. (b) Find the displacement over the given interval. (c) Find the distance travelled over then given interval. (d) Find the average velocity on [0, 5]. 7. Suppose that object A is located at position s = 0 at time t = 0. Object A starts moving along the s-axis with a velocity given by v(t) = 2at, where a is a positive constant. Object B is located at position s = c at time t = 0, where c is a positive constant. Object B starts moving along the s-axis with a constant velocity given by v(t) = b, where b is a positive constant. Show that A overtakes B at time: √ b + b2 + 4ac t= 2a 8. This question is worth 1 point. You are encouraged to work with your classmates! We want to know if you are collaborating and which websites you find helpful. Please answer the following: (a) Which classmates (if any) did you work with? Please indicate if they are in your recitation. (b) Which instructors did you talk to? (c) Were there any other people you talked to? (d) Which websites did you consult?

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