Chapter 5 | Integration
531
∑ i =1 n
∑ i =1 n
i 2
f ( x i )Δ x = 8 n 3
⎡ ⎣ n ( n +1)(2 n +1) 6 ⎡ ⎣ 2 n 3 +3 n 2 + n 6 ⎤ ⎦
⎤ ⎦
= 8
n 3
= 8
n 3
3 +24 n 2 +8 n 6 n 3
= 16 n
4 n + 8
= 8 3 +
.
6 n 2
Now, to calculate the definite integral, we need to take the limit as n →∞. We get ∫ 0 2 x 2 dx = lim n →∞ ∑ i =1 n f ( x i )Δ x
⎛ ⎝ 8 ⎛ ⎝ 8 3
⎞ ⎠
4 n + 8
= lim n →∞ = lim n →∞
3 +
6 n 2
⎛ ⎝ 8
⎞ ⎠
⎞ ⎠ + lim n →∞ ⎛
⎞ ⎠ + lim n →∞
⎝ 4 n
6 n 2
8 3 .
= 8 3 +0+0=
Use the definition of the definite integral to evaluate ∫ 0 3
5.7
(2 x −1) dx . Use a right-endpoint approximation
to generate the Riemann sum.
Evaluating Definite Integrals Evaluating definite integrals this way can be quite tedious because of the complexity of the calculations. Later in this chapter we develop techniques for evaluating definite integrals without taking limits of Riemann sums. However, for now, we can rely on the fact that definite integrals represent the area under the curve, and we can evaluate definite integrals by using geometric formulas to calculate that area. We do this to confirm that definite integrals do, indeed, represent areas, so we can then discuss what to do in the case of a curve of a function dropping below the x -axis. Example 5.8 Using Geometric Formulas to Calculate Definite Integrals
Use the formula for the area of a circle to evaluate ∫ 3 6
9−( x −3) 2 dx .
Solution The function describes a semicircle with radius 3. To find
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