Calculus III 14 Exam 1
Contents
Chapter 14 Exam
From Calculus 10e by Larson and Edwards, p. 1034. Exercises 3, 9, 13, 15, 17.
Exercise 3 Evaluating an Iterated Integral
Evaluate the iterated integral.
- $$\int_{0}^{1} \int_{0}^{1+x} (3x+2y) \: dy \: dx$$
Solution
$$\int_{0}^{1} \int_{0}^{1+x} (3x+2y) \: dy \: dx$$ | $$= \int_{0}^{1} \int_{0}^{1+x} 3x \: dy+ \int_{0}^{1+x} 2y \: dy \: dx $$ | Apply the Sum Rule |
$$= \int_{0}^{1} \left. \vphantom{\frac{3}{4}} 3xy \right]_{0}^{1+x} + \left. \vphantom{\frac{3}{4}} y^{2} \right]_{0}^{1+x} \: dx $$ | ||
$$= \int_{0}^{1} (x+1)(4x+1) \: dx$$ | ||
$$= \left. \vphantom{\frac{3}{4}} 2x^{3}- \frac{2}{3} x^{3} +2x^{2} + \frac{1}{2} x^{2} \right]_{0}^{1} = \frac{29}{6}$$ |
Exercise 9 Finding the Area for a Region
Use an iterated integral to find the area for the region bounded by the graphs.
- \(y=x, \: y=2x+2, \: x=0, \: x=4\)
Solution
The graph is shown in Figure 1. The equation for the area is
Evaluating the integral produces
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Exercise 13 Switching the Integration Order
Sketch the region \(R\) whose area is given by the iterated integral. Then switch the integration order and show that both orders yield the same area.
- $$ \int_{0}^{4} \int_{2x}^{8} \: dy \: dx$$
Solution
The sketch is shown in Figure 2. The first integration order, \(dy \: dx\), produces
The second integration order, \(dx \: dy\), requires new bounds, \(0 \leqslant x \leqslant 1/2y\) and \(0 \leqslant y \leqslant 8 \), which produces
The areas are equal. |
Exercise 15 Evaluating a Double Integral
Set up integrals for both integration orders. Use the more convenient order to evaluate the integral over the region \(R\).
- $$ \int_{R} \int 4xy \: dA$$
- \(R\): rectangle with vertices (0,0), (0,4), (2,4), (2,0).
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Solution The graph is shown in Figure 3. The bounds are \(0 \leqslant x \leqslant 2\) and \(0 \leqslant y \leqslant 4\). The integral for the area are
Since both are equally easy to solve the first will be used.
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Exercise 17 Finding Volume
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Use a double integral to find the volume for the solid in Figure 4.
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Parent Article: Calculus III Advanced (Course)