Think About It The graph of f consists of line segments and a semicircle, as shown in the figure. Evaluate each definite integral by using geometric formulas. (a) ∫ 0 2 f ( x ) d x (b) ∫ 2 6 f ( x ) d x (c) ∫ − 4 2 f ( x ) d x (d) ∫ − 4 6 f ( x ) d x (e) ∫ − 4 6 | f ( x ) | d x (f) ∫ − 4 6 [ f ( x ) + 2 ] d x
Think About It The graph of f consists of line segments and a semicircle, as shown in the figure. Evaluate each definite integral by using geometric formulas. (a) ∫ 0 2 f ( x ) d x (b) ∫ 2 6 f ( x ) d x (c) ∫ − 4 2 f ( x ) d x (d) ∫ − 4 6 f ( x ) d x (e) ∫ − 4 6 | f ( x ) | d x (f) ∫ − 4 6 [ f ( x ) + 2 ] d x
Solution Summary: The author explains how to calculate the integral displaystyleint_02f(x)dx.
Think About It The graph of f consists of line segments and a semicircle, as shown in the figure. Evaluate each definite integral by using geometric formulas.
(a)
∫
0
2
f
(
x
)
d
x
(b)
∫
2
6
f
(
x
)
d
x
(c)
∫
−
4
2
f
(
x
)
d
x
(d)
∫
−
4
6
f
(
x
)
d
x
(e)
∫
−
4
6
|
f
(
x
)
|
d
x
(f)
∫
−
4
6
[
f
(
x
)
+
2
]
d
x
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.
Think About It Use a graphing utility to graphthe functions f(x) = √x and g(x) = 6 arctan x. Forx > 0, it appears that g > f. Explain how you knowthat there exists a positive real number a such thatg < f for x > a. Approximate the number a.
Evaluate the function shown using the Fundamental Theorem of Calculus.Check your work by evaluating the integral using geometry.
Sketch and use geometry (not a Riemann sum or the Fundamental Theorem of Calculus) to evaluate the following definite integrals. Show work, thank you! Part a and b.
Chapter 5 Solutions
Calculus: Early Transcendental Functions (MindTap Course List)
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