EXAMPLE 4 Evaluate the following integrals by interpreting each in terms of areas.(a)    39 − x2 dx0(b)    10(x − 3) dx0SOLUTION (a) Since f(x) = 9 − x2 ≥ 0, we can interpret this integral as the area under the curve y = 9 − x2 from 0 to . But since y2 = , we get x2 + y2 = 9, which shows that the graph of f is a quarter-circle with radius in the top figure. Therefore,39 − x2 dx0 = 14𝜋(3)2 = .(b) The graph of y = x − 3 is the line with slope shown in the bottom figure. We compute the integral as the difference of the areas of the two triangles:10(x − 3) dx0 = A1 − A2 = − 4.5 = .

The graph of f is shown. Evaluate each integral by interpreting it in terms of areas.(a)    10f(x) dx0 (b)    25f(x) dx0 (c)    35f(x) dx25 (d)    45f(x) dx0

(b) The graph of y = x − 3 is the line with slope shown in the bottom figure. We compute the integral as the difference of the areas of the two triangles:10(x − 3) dx0 = A1 − A2 = − 4.5 = .

To find the area between the curves \(f(x) = x\) and \(f(x) = x^2\) in the interval \(0 \leq x \leq 1\), follow these steps: 1. **Set up the integral**: The area between the curves is given by the integral of the difference between the two functions over the given interval: \[ \int_0^1 (x - x^2) \, dx \] 2. **Evaluate the integral**: \[ \int_0^1 (x - x^2) \, dx = \int_0^1 x \, dx - \int_0^1 x^2 \, dx \] 3. **Find the antiderivatives**: \[ \int x \, dx = \frac{x^2}{2} + C \] \[ \int x^2 \, dx = \frac{x^3}{3} + C \] 4. **Apply the limits of integration**: \[ \left[ \frac{x^2}{2} \right]_0^1 - \left[ \frac{x^3}{3} \right]_0^1 \] 5. **Evaluate the definite integrals**: \[ \left[ \frac{x^2}{2} \right]_0^1 = \frac{1^2}{2} - \frac{0^2}{2} = \frac{1}{2} \] \[ \left[ \frac{x^3}{3} \right]_0^1 = \frac{1^3}{3} - \frac{0^3}{3} = \frac{1}{3} \] 6. **Subtract the results**: \[ \frac{1}{2} - \frac{1}{3} = \frac{3}{6} - \frac{2}{6} = \frac{1}{6} \] So, the area between the curves \(f(x) = x\) and \(f(x) = x^2\) in the interval \(0 \leq x \leq 1\) is \(\frac{1}{6}\). The correct answer is: - \(\frac{1}{6}\)

Sketch the graph of the function 𝑥3:(a) Find the signed area between the curve, the 𝑥-axis, and the lines 𝑥=−2 and 𝑥=1.Signed area = (b) Find the geometric area between the curve, the 𝑥-axis, and the lines 𝑥=−2 and 𝑥=1.True area =

Use left and right endpoints and the given number of rectangles to find two approximations of the area of the region between the graph of the function and the x-axis over the given interval.f(x) = 2x + 9,   [0, 2],   4 rectangles