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Find the exact value of the integral using formulas from geometry. 6 Se 2 (3 + x)dx

Find the exact value of the integral using formulas from geometry. 6 Se 2 (3 + x)dx

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter8: Further Techniques And Applications Of Integration
Section8.CR: Chapter 8 Review
Problem 14CR
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**Question:** Find the exact value of the integral using formulas from geometry. \[ \int_{2}^{6} (3 + x) \, dx \] **Explanation:** To solve this problem using geometry, we need to interpret the integral as the area under the function \(3 + x\) between \(x = 2\) and \(x = 6\). 1. **Graph and Shape Identification:** - The function \(3 + x\) represents a straight line with a slope of 1 and a y-intercept of 3. - For \(x = 2\), the function value is \(3 + 2 = 5\). - For \(x = 6\), the function value is \(3 + 6 = 9\). - The integral \(\int_{2}^{6} (3 + x) \, dx\) represents the area under this line from \(x = 2\) to \(x = 6\). - The area under the line forms a trapezoid. 2. **Trapezoid Area Calculation:** - The trapezoid has parallel sides (bases) of lengths 5 and 9 (values of the function at \(x = 2\) and \(x = 6\) respectively). - The height (distance between \(x = 2\) and \(x = 6\)) is \(6 - 2 = 4\). The area \(A\) of a trapezoid can be calculated using the formula: \[ A = \frac{1}{2} \times ( \text{base}_1 + \text{base}_2 ) \times \text{height} \] Substituting the values we have: \[ A = \frac{1}{2} \times (5 + 9) \times 4 = \frac{1}{2} \times 14 \times 4 = 28 \] Thus, the exact value of the integral \(\int_{2}^{6} (3 + x) \, dx\) is 28.
Transcribed Image Text:**Question:** Find the exact value of the integral using formulas from geometry. \[ \int_{2}^{6} (3 + x) \, dx \] **Explanation:** To solve this problem using geometry, we need to interpret the integral as the area under the function \(3 + x\) between \(x = 2\) and \(x = 6\). 1. **Graph and Shape Identification:** - The function \(3 + x\) represents a straight line with a slope of 1 and a y-intercept of 3. - For \(x = 2\), the function value is \(3 + 2 = 5\). - For \(x = 6\), the function value is \(3 + 6 = 9\). - The integral \(\int_{2}^{6} (3 + x) \, dx\) represents the area under this line from \(x = 2\) to \(x = 6\). - The area under the line forms a trapezoid. 2. **Trapezoid Area Calculation:** - The trapezoid has parallel sides (bases) of lengths 5 and 9 (values of the function at \(x = 2\) and \(x = 6\) respectively). - The height (distance between \(x = 2\) and \(x = 6\)) is \(6 - 2 = 4\). The area \(A\) of a trapezoid can be calculated using the formula: \[ A = \frac{1}{2} \times ( \text{base}_1 + \text{base}_2 ) \times \text{height} \] Substituting the values we have: \[ A = \frac{1}{2} \times (5 + 9) \times 4 = \frac{1}{2} \times 14 \times 4 = 28 \] Thus, the exact value of the integral \(\int_{2}^{6} (3 + x) \, dx\) is 28.
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