Question

Find the area of the indicated region. We suggest you graph the curves to check whether one is above the other or whether they cross, and that you use technology to check your answer. HINT [See Example 1.] Between y = x2 and y = −3 for x in [−1, 1]

Answer 1

$y_1 = x^2 A_1 = \int\limits^1_{-1} {x^2} \, dx = \frac{1}{3} x^3 \Big|^{+1}_{-1} = \frac{2}{3} y_2 = -3 A_2 = \int\limits^1_{-1} {-3} \, dx = -3x \Big|_{-1}^{+1} = -6$

Answer 2

The area of the indicated region is 20/3 unit^2

Data;

• y = x^2
• y = -3
• x = [-1, 1]

Area Under the Curve

To find the area under the curve, we have to integrate through the sides

$A = \int\limits^x^=^1_x_=_-_1 [{\int\limits^x^2_y_=_-_3 {x} \, dy } \,] dx$

This becomes

$A = \int\limits^x^=^1_x_=_-_1 {(x^2+3)} \, dx$

resolving this,

$A = \frac{20}{3}$

The area of the indicated region is 20/3 unit^2

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