Please answer the question shown below and show all work! Thank you! **Educational Content: Calculating Area Using Partial Fractions****Problem Statement:**Use the method of partial fractions to find the area bounded by the curve given by the equation:\[ y = \frac{2x - 8}{x^2 - 5x + 6} \]The area is bounded by the x-axis, \( x = 9 \), and \( x = 13 \). Show all work.**Solution Explanation:**To solve this problem, we follow these steps:1. **Identify the Fraction to Decompose:** The function is in the form of a rational function \( \frac{2x - 8}{x^2 - 5x + 6} \).2. **Factor the Denominator:** The quadratic denominator \( x^2 - 5x + 6 \) can be factored into \((x - 2)(x - 3)\).3. **Setup Partial Fractions:** The expression can be decomposed as: \[ \frac{2x - 8}{(x - 2)(x - 3)} = \frac{A}{x - 2} + \frac{B}{x - 3} \] Here, \( A \) and \( B \) are constants we need to determine.4. **Solve for Constants \( A \) and \( B \):** Multiply through by the common denominator to clear fractions: \[ 2x - 8 = A(x - 3) + B(x - 2) \] Expand and equate coefficients to solve for \( A \) and \( B \).5. **Integrate to Find the Area:** Once the partial fractions are found, integrate each fraction separately between the limits \( x = 9 \) and \( x = 13 \) to find the bounded area.6. **Apply Calculus to Find Definite Integrals:** Use techniques of integration to evaluate the area under the curve within the specified limits.**Conclusion:**By using partial fractions, we break down complex rational expressions into simpler forms, making it easier to integrate. This method is particularly useful in finding areas under curves, providing a systematic approach to solving complex calculus problems.

Name: Please answer the question shown below and show all work! Thank you! *
Uploaded: 2024-03-20T07:07:37.000Z
Channel: Bartleby
Description: Please answer the question shown below and show all work! Thank you! **Educational Content: Calculating Area Using Partial Fractions****Problem Statement:**Use the method of partial fractions to find the area bounded by the curve given by the equatio

The given function y -y = 2x-8x2-5x+6y = 2x-8x2-2x-3x+6y = 2x-8x(x-2)-3(x-2)y = 2x-8x-2x-3Using Partial Fraction - 2x-8x-2x-3 = Ax-2+Bx-32x-8x-2x-3 = Ax-3+Bx-2x-2x-32x-8 = Ax-3+Bx-2We have to find the value of A and B,Put x = 3,23-8 = A3-3+B(3-2)B = -2Put x = 2,2(2)-8 = A(2-3)+B(2-2)-A = -4A = 4Then,y = 2x-8x-2x-3 = 4x-2-2x-3 .............(1)Area under the curve y between x=9 and x=13- Area = ∫913ydxArea = ∫9134x-2-2x-3dxArea = 4∫913dxx-2-2∫913dxx-3Area = 4ln(x-2)913-2lnx-3913Area = 4ln13-2-ln9-2-2ln13-3-ln9-3Area = 4ln11-ln7-2ln10-ln6Area = 4ln117-2ln106Area = ln1174-ln532Area = ln146412401-ln259Area = ln146412401259Area = ln(2.195)Hence area under the curve from x=9 and x=13 is ln(2.195) square unit.