6 Evaluate the integral from geometry whose (signed) sketch the area is by region represented by the definite integral using a geomety formule. a) 9-x² dx ) -4 dx له (....
6 Evaluate the integral from geometry whose (signed) sketch the area is by region represented by the definite integral using a geomety formule. a) 9-x² dx ) -4 dx له (....
Advanced Engineering Mathematics
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Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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![**Problem 5: Evaluating Definite Integrals Using Geometric Methods**
Evaluate the integral from geometry. Sketch the region whose (signed) area is represented by the definite integral by using a geometry formula.
---
**a)** \[
\int_{-3}^{0} \sqrt{9 - x^2} \, dx
\]
**b)** \[
\int_{-3}^{0} -4 \, dx
\]
**c)** \[
\int_{0}^{4} (2x - 4) \, dx
\]
---
**Explanation of Geometric Interpretation**
- **a)** The integral \(\int_{-3}^{0} \sqrt{9 - x^2} \, dx\) represents the area of a semicircle above the x-axis with radius 3, centered at the origin, evaluated from x = -3 to x = 0.
- **b)** The integral \(\int_{-3}^{0} -4 \, dx\) evaluates the area of a rectangle with height -4 and width 3, from x = -3 to x = 0.
- **c)** The integral \(\int_{0}^{4} (2x - 4) \, dx\) represents the area under the straight line \(y = 2x - 4\) from x = 0 to x = 4.
**Sketching Solutions**
1. **a)** Sketch a semicircle in the coordinate plane, with its center at the origin and a radius of 3 units. Shade the left half from x = -3 to x = 0.
2. **b)** Sketch a rectangle below the x-axis, spanning from x = -3 to x = 0, with a constant height of -4 units.
3. **c)** Sketch the area under the line \(y = 2x - 4\), starting at the point (0, -4) and ending at the point (4, 4), then find the area between the line and the x-axis.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fee8bddee-32fd-433e-8085-8fa50d25b5ad%2F988bb351-cd03-4bad-88df-c932b91942ef%2F2gsno9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 5: Evaluating Definite Integrals Using Geometric Methods**
Evaluate the integral from geometry. Sketch the region whose (signed) area is represented by the definite integral by using a geometry formula.
---
**a)** \[
\int_{-3}^{0} \sqrt{9 - x^2} \, dx
\]
**b)** \[
\int_{-3}^{0} -4 \, dx
\]
**c)** \[
\int_{0}^{4} (2x - 4) \, dx
\]
---
**Explanation of Geometric Interpretation**
- **a)** The integral \(\int_{-3}^{0} \sqrt{9 - x^2} \, dx\) represents the area of a semicircle above the x-axis with radius 3, centered at the origin, evaluated from x = -3 to x = 0.
- **b)** The integral \(\int_{-3}^{0} -4 \, dx\) evaluates the area of a rectangle with height -4 and width 3, from x = -3 to x = 0.
- **c)** The integral \(\int_{0}^{4} (2x - 4) \, dx\) represents the area under the straight line \(y = 2x - 4\) from x = 0 to x = 4.
**Sketching Solutions**
1. **a)** Sketch a semicircle in the coordinate plane, with its center at the origin and a radius of 3 units. Shade the left half from x = -3 to x = 0.
2. **b)** Sketch a rectangle below the x-axis, spanning from x = -3 to x = 0, with a constant height of -4 units.
3. **c)** Sketch the area under the line \(y = 2x - 4\), starting at the point (0, -4) and ending at the point (4, 4), then find the area between the line and the x-axis.
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