Changing the order of integration Rewrite the following integrals using the indicated order of integration and then evaluate the resulting integral. 42. ∫ 0 4 ∫ 0 16 − x 2 ∫ 0 16 − x 2 − z 2 d y d z d x in the order dz dy dx
Changing the order of integration Rewrite the following integrals using the indicated order of integration and then evaluate the resulting integral. 42. ∫ 0 4 ∫ 0 16 − x 2 ∫ 0 16 − x 2 − z 2 d y d z d x in the order dz dy dx
Changing the order of integrationRewrite the following integrals using the indicated order of integration and then evaluate the resulting integral.
42.
∫
0
4
∫
0
16
−
x
2
∫
0
16
−
x
2
−
z
2
d
y
d
z
d
x
in the order dz dy dx
With differentiation, one of the major concepts of calculus. Integration involves the calculation of an integral, which is useful to find many quantities such as areas, volumes, and displacement.
Evaluate the definite integral. Use the integration capabilities of a graphing utility to verify your result.
2x
dx
Vx2 + 16
Step 1
2x
To find the definite integral
Vx +
dx, apply the basic integration rule.
+ 16
Let u = x² + 16. Differentiate u in terms of x, du =
A 2 x dx.
Step 2
Rewrite the integration in terms of u.
2x
x = 3
du
V x2 + 16
= xp
Step 3
Apply the power rule of integration.
rx = 3
x = 3
du =
du
+1 1x = 3
+1Jo
1x = 3
Apply integration by parts twice for evaluation the following integrals.
Evaluate the definite integral. Use the integration capabilities of a graphing utility to verify your result.
2x
x² + 16
xp
Step 1
2x
To find the definite integral
Vx² +
dx, apply the basic integration rule.
Let u = x² + 16. Differentiate u in terms of x, du =
2 x dx.
Step 2
Rewrite the integration in terms of u.
2x
dx =
du
x² + 16
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