Chapter 5 | Integration
563
169. [T] ⌠ ⌡
162. The graph of y = ∫ 0 x linear function, is shown here.
4 x 2
ℓ ( t ) dt , where ℓ is a piecewise
dx over [1, 4]
In the following exercises, evaluate each definite integral using the Fundamental Theorem of Calculus, Part 2. 170. ∫ −1 2 ⎛ ⎝ x 2 −3 x ⎞ ⎠ dx 171. ∫ −2 3 ⎛ ⎝ x 2 +3 x −5 ⎞ ⎠ dx 172. ∫ −2 3 ( t +2)( t −3) dt 173. ∫ 2 3 ⎛ ⎝ t 2 −9 ⎞ ⎠ ⎛ ⎝ 4− t 2 ⎞ ⎠ dt
a. Over which intervals is ℓ positive? Over which intervals is it negative? Over which, if any, is it zero? b. Over which intervals is ℓ increasing? Over which is it decreasing? Over which, if any, is it constant? c. What is the average value of ℓ ? 163. The graph of y = ∫ 0 x ℓ ( t ) dt , where ℓ is a piecewise linear function, is shown here.
2
174. ∫
x 9 dx
1
1
175. ∫
x 99 dx
0
8 ⎛ ⎝ 4 t 5/2 −3 t 3/2 ⎞
176. ∫
⎠ dt
4
4 ⎛
⎞ ⎠ dx
177. ⌠
⎝ x 2 − 1 x 2
⌡ 1/4
a. Over which intervals is ℓ positive? Over which intervals is it negative? Over which, if any, is it zero? b. Over which intervals is ℓ increasing? Over which is it decreasing? Over which intervals, if any, is it constant? c. What is the average value of ℓ ? In the following exercises, use a calculator to estimate the area under the curve by computing T 10 , the average of the left- and right-endpoint Riemann sums using N =10 rectangles. Then, using the Fundamental Theorem of Calculus, Part 2, determine the exact area. 164. [T] y = x 2 over [0, 4] 165. [T] y = x 3 +6 x 2 + x −5 over [−4, 2] 166. [T] y = x 3 over ⎡ ⎣ 0, 6 ⎤ ⎦ 167. [T] y = x + x 2 over [1, 9] 168. [T] ∫ (cos x −sin x ) dx over [0, π ]
⌡ 1 2
178. ⌠
2 x 3
dx
4 1
179. ⌠
dx
⌡ 1
2 x
⌡ 1 4
180. ⌠
2− t t 2
dt
16
181. ⌠
dt t 1/4
⌡ 1
2 π
182. ∫
cos θdθ
0
π /2
183. ∫
sin θdθ
0
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