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$The diagram shows the curve y=x^(2)-3x+1 and the line y=5. Find the area of the shaded region. Give your answer as a fraction in its simplest form.$

The diagram shows the curve $y=x^{2}-3 x+1$ and the line $y=5$ . $\newline$ Find the area of the shaded region. Give your answer as a fraction in its simplest form.

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Math Problems

Algebra 2

Write a quadratic function in vertex form

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Q. The diagram shows the curve

y=x^{2}-3 x+1

and the line

y=5

\newline

Find the area of the shaded region. Give your answer as a fraction in its simplest form.

Set Intersection Points: To find the area of the shaded region, we need to calculate the definite integral of the difference between the line $y=5$ and the curve $y=x^2-3x+1$ from the left intersection point to the right intersection point.
Calculate Integral: First, find the points of intersection by setting the equations equal to each other: $x^2 - 3x + 1 = 5$ .
Find Roots: Solve the equation $x^2 - 3x + 1 = 5$ by subtracting $5$ from both sides to get $x^2 - 3x - 4 = 0$ .
Set Up Integral: Factor the quadratic equation: $(x - 4)(x + 1) = 0$ .
Integrate Function: Find the roots of the equation by setting each factor equal to zero: $x - 4 = 0$ and $x + 1 = 0$ , which gives us $x = 4$ and $x = -1$ .
Evaluate Integral: The points of intersection are $x = -1$ and $x = 4$ . Now, set up the integral from $-1$ to $4$ of the function $5 - (x^2 - 3x + 1)$ .
Upper Limit Calculation: The integral becomes $\int_{-1}^{4} (4 - x^2 + 3x) \, dx$ .
Lower Limit Calculation: Integrate the function: $\int (4 - x^2 + 3x) \, dx = 4x - \frac{1}{3}x^3 + \frac{3}{2}x^2$ .
Subtract Values: Evaluate the integral from $-1$ to $4$ : $[4x - (1/3)x^3 + (3/2)x^2]$ from $-1$ to $4$ .
Subtract Values: Evaluate the integral from $-1$ to $4$ : $[4x - (1/3)x^3 + (3/2)x^2]$ from $-1$ to $4$ .Plug in the upper limit of the integral: $4(4) - (1/3)(4)^3 + (3/2)(4)^2 = 16 - (1/3)(64) + (3/2)(16)$ .
Subtract Values: Evaluate the integral from $-1$ to $4$ : $[4x - (1/3)x^3 + (3/2)x^2]$ from $-1$ to $4$ .Plug in the upper limit of the integral: $4(4) - (1/3)(4)^3 + (3/2)(4)^2 = 16 - (1/3)(64) + (3/2)(16)$ .Calculate the value for the upper limit: $16 - 64/3 + 24 = 48/3 - 64/3 + 72/3 = 56/3$ .
Subtract Values: Evaluate the integral from $-1$ to $4$ : $[4x - (1/3)x^3 + (3/2)x^2]$ from $-1$ to $4$ .Plug in the upper limit of the integral: $4(4) - (1/3)(4)^3 + (3/2)(4)^2 = 16 - (1/3)(64) + (3/2)(16)$ .Calculate the value for the upper limit: $16 - 64/3 + 24 = 48/3 - 64/3 + 72/3 = 56/3$ .Plug in the lower limit of the integral: $4(-1) - (1/3)(-1)^3 + (3/2)(-1)^2 = -4 - (1/3)(-1) + (3/2)(1)$ .
Subtract Values: Evaluate the integral from $-1$ to $4$ : $[4x - (1/3)x^3 + (3/2)x^2]$ from $-1$ to $4$ .Plug in the upper limit of the integral: $4(4) - (1/3)(4)^3 + (3/2)(4)^2 = 16 - (1/3)(64) + (3/2)(16)$ .Calculate the value for the upper limit: $16 - 64/3 + 24 = 48/3 - 64/3 + 72/3 = 56/3$ .Plug in the lower limit of the integral: $4(-1) - (1/3)(-1)^3 + (3/2)(-1)^2 = -4 - (1/3)(-1) + (3/2)(1)$ .Calculate the value for the lower limit: $-4 + 1/3 + 3/2 = -12/3 + 1/3 + 4.5/3 = -6.5/3$ .
Subtract Values: Evaluate the integral from $-1$ to $4$ : $[4x - (1/3)x^3 + (3/2)x^2]$ from $-1$ to $4$ .Plug in the upper limit of the integral: $4(4) - (1/3)(4)^3 + (3/2)(4)^2 = 16 - (1/3)(64) + (3/2)(16)$ .Calculate the value for the upper limit: $16 - 64/3 + 24 = 48/3 - 64/3 + 72/3 = 56/3$ .Plug in the lower limit of the integral: $4(-1) - (1/3)(-1)^3 + (3/2)(-1)^2 = -4 - (1/3)(-1) + (3/2)(1)$ .Calculate the value for the lower limit: $-4 + 1/3 + 3/2 = -12/3 + 1/3 + 4.5/3 = -6.5/3$ .Subtract the lower limit value from the upper limit value: $56/3 - (-6.5/3) = 56/3 + 6.5/3 = 62.5/3$ .