21-121  IM/ECON Calculus  Fall 2000

Review Problems for Test 4

1. Assume that ∫_{0}^{5}f(x)dx = −7, ∫_{0}^{1}f(x)dx = 2, and ∫_{1}^{5}g(x)dx = 6.

(a) Find ∫_{1}^{5}f(x)dx.

(b) Find ∫_{1}^{5}(3f(x) − 4g(x))dx.

2. Determine each of the following:

(a) \frac{d}{dx}∫_{lnx}^{x^{2}} \sqrt{1 + t^{4}} dt

(b) \frac{d}{dx}∫_{x}^{3}e^{t^{3}} dt

3. Evaluate each of the following indefinite integrals (a) ∫ (x^{2} + x^{3} − 3x^{4}) dx

(b) ∫ (e^{3x} + e^{−2x}) dx

(c) ∫ \sqrt{x}(x + 1)^{2}dx

(d) ∫ e^{−x} (e^{2x} + e^{5x}) dx

4. Evaluate each of the following definite integrals (a) ∫_{0}^{1}(x^{3} − 6x^{2} + 1)dx

(b) ∫_{4}^{9} \frac{dz}{\sqrt{z}}

(c) ∫_{0}^{ln^{3}} (\frac{e^{x} + e^{3x}}{e^{2x}} ) dx

5. Evaluate each of the following indefinite integrals.

(a) ∫ \frac{x^{3}}{(1 − x^{4})^{6}} dx

(b) ∫ \frac{x^{2}}{1 + x^{3}} dx

(c) ∫ \frac{dx}{x lnx}

(d) ∫ x^{3}lnxdx

(e) ∫ x^{2}\sqrt{x − 1} dx

(f) ∫ \sqrt{1 + \sqrt{x}} dx

6. Evaluate each of the following definite integrals

(a) ∫_{0}^{6}\sqrt{2x + 4} dx

(b) ∫_{2}^{3} \frac{dx}{x(lnx)^{2}}

(c) ∫_{0}^{1}xe^{2x}dx

(d) ∫_{0}^{1} \frac{dx}{1 + \sqrt{x}}

7. Given that f satisfied f′(x) = f(x)^{2} for all x ∈ [0, 1] and f(0) = −1, f(1) = −\frac{1}{2}, find ∫_{0}^{1}f(x)^{4}dx.

8. Evaluate each limit by identifying it with a definite integral.

(a) lim_{n→∞} \frac{1}{n} ∑_{i=1}^{n}e^{\frac{i}{n}}

(b) lim_{n→∞} \frac{3}{n} ∑_{i=1}^{n} \sqrt{1 + \frac{3i}{n}}

9. Find the area of (a) the region bounded by y = x + 1 and y = x^{2} − 1.

(b) the region bounded by y = 0, y = lnx, and x = e.

10. Find the volume generated when

(a) the region bounded by y = 0, y = e^{3x}, x = 0, and x = 1 is rotated about the x-axis.

(b) the region bounded by y = \sqrt{x}, y = 2, and x = 0 is rotated about the y-axis.