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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) (b) 4 (C) (d) (f) (-4,-1) [²₁ f(x) dx forex f(x) dx f(x) dx Lor 1₁ X X f(x) dx [o [r(x). |f(x) dx [f(x) + 2] dx 1 -1 (4,2) 3 4 5 6

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) (b) 4 (C) (d) (f) (-4,-1) [²₁ f(x) dx forex f(x) dx f(x) dx Lor 1₁ X X f(x) dx [o [r(x). |f(x) dx [f(x) + 2] dx 1 -1 (4,2) 3 4 5 6

Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
18th Edition
ISBN:9780079039897
Author:Carter
Publisher:Carter
Chapter1: Expressions And Functions
Section1.8: Interpreting Graphs Of Functions
Problem 1GP
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All the solutions are incorrect except for (b) =4

The graph of \( f \) consists of line segments and a semicircle. ### Graph Analysis The graph is defined on the x-axis from \(-4\) to \(6\). There are: - Line segments starting at \((-4, -1)\) to \((0, 0)\). - A semicircular curve is present between \(0\) and \(4\) with a peak at \(y = 2\) and center at \((2, 0)\). - A line segment from \((4, 2)\) down to \((6, 0)\). ### Evaluate each definite integral using geometric formulas: (a) \(\int_{0}^{2} f(x) \, dx\) (b) \(\int_{2}^{6} f(x) \, dx\) - Solution: \(4\) (c) \(\int_{-4}^{2} f(x) \, dx\) (d) \(\int_{-4}^{6} f(x) \, dx\) (e) \(\int_{-4}^{6} |f(x)| \, dx\) (f) \(\int_{-4}^{6} [f(x) + 2] \, dx\) #### Geometrical Explanation: The semicircle has a radius of 2, with its area computed as \(\frac{1}{2} \pi r^2\). The areas under and above the x-axis are geometrically simple shapes (triangles, rectangles, semicircles) whose areas can be calculated using basic geometric formulas. Please replace the boxes with the calculated areas, considering parts under and above the x-axis, positive and negative areas respectively.
Transcribed Image Text:The graph of \( f \) consists of line segments and a semicircle. ### Graph Analysis The graph is defined on the x-axis from \(-4\) to \(6\). There are: - Line segments starting at \((-4, -1)\) to \((0, 0)\). - A semicircular curve is present between \(0\) and \(4\) with a peak at \(y = 2\) and center at \((2, 0)\). - A line segment from \((4, 2)\) down to \((6, 0)\). ### Evaluate each definite integral using geometric formulas: (a) \(\int_{0}^{2} f(x) \, dx\) (b) \(\int_{2}^{6} f(x) \, dx\) - Solution: \(4\) (c) \(\int_{-4}^{2} f(x) \, dx\) (d) \(\int_{-4}^{6} f(x) \, dx\) (e) \(\int_{-4}^{6} |f(x)| \, dx\) (f) \(\int_{-4}^{6} [f(x) + 2] \, dx\) #### Geometrical Explanation: The semicircle has a radius of 2, with its area computed as \(\frac{1}{2} \pi r^2\). The areas under and above the x-axis are geometrically simple shapes (triangles, rectangles, semicircles) whose areas can be calculated using basic geometric formulas. Please replace the boxes with the calculated areas, considering parts under and above the x-axis, positive and negative areas respectively.
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