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
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fab6702de-5579-4c14-bacb-4d8e7e421acd%2Fbc32942f-d83c-4ce3-becb-063cbc06b513%2Fbw300g_processed.png&w=3840&q=75)
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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