Calculus Test 3

Instructor: Prof. A. Mingarelli
Mar. 1997

NOTATION: Pi = 3.14159... is our usual Greek number

1. Find the volume of the solid of revolution obtained by rotating the region bounded by the curve y = sqrt(x+2), y >= 0, x = 2, and x = 7 about the y-axis. Do this in 2 steps:

Solution: Part 1: Use a vertical fibre: Its extremities will have the coordinates (x,0) and (x, sqrt(x+2)). We see that rin ~ x, rout ~ x and its height = sqrt(x+2). Next, rout - rin = dx. So, its volume (as a disk, or washer with no inner radius) is given by dV = 2 Pi x sqrt(x + 2) dx and the integral for the full volume is given by the integral of the function f(x) = 2 Pi x sqrt(x + 2) between the limits x = 2 and x = 7.

Part 2: Integrate by parts using: u = x, dv = sqrt(x+2). Then du = dx and v = (2/3) (x+2)3/2. The resulting integral is easily evaluated (but may be frustrating) and leads to an answer of 1772 Pi/ 15 ~ 371.13 .

2. Find the area between the curves y = sin(x) and y = cos(x) for 0 <= x <= Pi/4 .

Solution:Note that on this interval, cos(x) exceeds sin(x) and so the area of a typical vertical fibre is given by (cos(x) - sin(x)) dx, right? (Recall that its coordinates are given by (x, sin(x)) and (x, cos(x))). Thus its area is given by the integral of cos(x) - sin(x) between the limits x = 0 and x = Pi/4.... This gives the value: sqrt(2) - 1 ~ 0.41425...

3. Evaluate the definite integral of the function f(x) = tan3(x) sec2(x) between the limits x = 0 and x = Pi/4.

Solution: Let u = tan(x), du = sec2(x) dx. Then we see immediately that tan4(x) /4 is an antiderivative of f and so we need to evaluate this between the limits x = 0 and x = Pi/4. This gives the value: 1/4.

4. Evaluate the definite integral of the function f(x) = 2 / (x2 + 1) between the limits x = 0 and x = 1 using Simpson's Rule with n = 4.

Solution: Here a = 0, b = 1, and n = 4. Furthermore, xo = 0, x1 = 1/4, ... Thus, the integral is given approximately by the expression: 2 (1/12) { (1.0000) + 4 (0.9412) + 2 (0.8000) + 4 (0.6400) + (0.5000) } = 2 (0.7854) = 1.5708.

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