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find the area of the figure bonded by $y=x^2-2x+5$ , $y=5x-5$ . use integral

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Complex conjugate theorem

Full solution

Q. find the area of the figure bonded by

y=x^2-2x+5

,

y=5x-5

. use integral

Find Intersection Points: First, we need to find the points of intersection between the two curves. Set the equations equal to each other: $x^2 - 2x + 5 = 5x - 5$ .
Solve for x: Solve for x: $x^2 - 2x + 5 - 5x + 5 = 0$ , which simplifies to $x^2 - 7x + 10 = 0$ .
Factor Quadratic Equation: Factor the quadratic equation: $(x - 5)(x - 2) = 0$ .
Find Roots: Find the roots: $x = 5$ and $x = 2$ . These are the points of intersection.
Set up Integral: Now, set up the integral to find the area between the curves from $x = 2$ to $x = 5$ . The area $A$ is given by the integral from $2$ to $5$ of $(5x - 5) - (x^2 - 2x + 5)$ dx.
Simplify Integrands: Simplify the integrand: $A = \int_{2}^{5} (5x - 5 - x^2 + 2x - 5) \, dx$ .
Combine Like Terms: Combine like terms: $A = \int_{2}^{5} (-x^2 + 7x - 10) \, dx$ .
Integrate Function: Integrate the function: $A = \left[\left(-\frac{1}{3}\right)x^3 + \left(\frac{7}{2}\right)x^2 - 10x\right]$ from $2$ to $5$ .
Upper Limit Calculation: Plug in the upper limit of integration: $A = (-\frac{1}{3})(5)^3 + (\frac{7}{2})(5)^2 - 10(5)$ .
Calculate Upper Limit: Calculate the value at the upper limit: $A = \left(-\frac{1}{3}\right)(125) + \left(\frac{7}{2}\right)(25) - 50$ .
Lower Limit Calculation: Simplify the calculation: $A = -41.67 + 87.5 - 50$ .
Calculate Lower Limit: Plug in the lower limit of integration: $A = [(-\frac{1}{3})(2)^3 + (\frac{7}{2})(2)^2 - 10(2)] - A$ .
Combine Values: Calculate the value at the lower limit: $A = [(-\frac{1}{3})(8) + (\frac{7}{2})(4) - 20] - A$ .
Final Area Calculation: Simplify the calculation: $A = -2.67 + 14 - 20 - A$ .
Final Area Calculation: Simplify the calculation: $A = -2.67 + 14 - 20 - A$ .Combine the values from the upper and lower limits: $A = (-41.67 + 87.5 - 50) - (-2.67 + 14 - 20)$ .
Final Area Calculation: Simplify the calculation: $A = -2.67 + 14 - 20 - A$ .Combine the values from the upper and lower limits: $A = (-41.67 + 87.5 - 50) - (-2.67 + 14 - 20)$ .Simplify the final area calculation: $A = -4.17 + 2.67 + 14 - 20$ .

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