Find the Area Between the Curves x=-2 , x=1 , y=2x , y=x^2-3

Problem

(∫_−2^1)(|2*x−(x2−3)|*d(x))

Solution

1. Identify the upper and lower functions on the interval [−2,1] by comparing their values.

2. Evaluate the functions at a test point within the interval, such as x=0

3. Determine that y=2*x is 0 and y=x2−3 is −3 at x=0 so 2*x≥x2−3 on this interval.

4. Set up the definite integral for the area using the formula (∫_a^b)((upper−lower)*d(x))

A=(∫_−2^1)((2*x−(x2−3))*d(x))

5. Simplify the integrand before integrating.

A=(∫_−2^1)((−x2+2*x+3)*d(x))

6. Integrate the expression term by term using the power rule.

(∫_^)((−x2+2*x+3)*d(x))=−(x3)/3+x2+3*x

7. Apply the Fundamental Theorem of Calculus by evaluating at the upper limit 1 and lower limit −2

A=(−((1)3)/3+(1)2+3*(1))−(−((−2)3)/3+(−2)2+3*(−2))

8. Calculate the values for each limit.

A=(−1/3+1+3)−(8/3+4−6)

9. Simplify the arithmetic to find the final area.

A=11/3−2/3

A=9/3

Final Answer

(∫_−2^1)((2*x−(x2−3))*d(x))=3

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