To solve the problems in the image, we'll calculate the area of the shaded regions using definite integrals. The problems seem to be related to finding the area under curves, which is a common application of integrals. I'll go through each part step by step.

Part a: $y = x^2 + 2$

We need to find the area under the curve $y = x^2 + 2$ from $x = 0$ to $x = 3$.

1. Set up the integral: $\text{Area} = \int_{0}^{3} (x^2 + 2) \, dx$

2. Compute the integral: $\int (x^2 + 2) \, dx = \frac{x^3}{3} + 2x$

3. Evaluate from $0$ to $3$: [ \text{Area} = \left[\frac{3^3}{3} + 2(3)\right] - \left[\frac{0^3}{3} + 2(0)\right] ] [ = \left[\frac{27}{3} + 6\right] - [0] = 9 + 6 = 15 ]

So, the area under the curve is 15 square units.

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Part b: $x = 4y - y^2$

This equation can be rewritten as $y^2 - 4y + x = 0$, but let's stick to the integral method to find the area under the curve.

1. Rewrite the curve to solve for $y$ (if needed): The given equation is $x = 4y - y^2$. We'll integrate with respect to $y$ between $y = 0$ and $y = 4$.

2. Set up the integral: $\text{Area} = \int_{0}^{4} (4y - y^2) \, dy$

3. Compute the integral: $\int (4y - y^2) \, dy = 2y^2 - \frac{y^3}{3}$

4. Evaluate from $0$ to $4$: [ \text{Area} = \left[2(4)^2 - \frac{(4)^3}{3}\right] - \left[2(0)^2 - \frac{(0)^3}{3}\right] ] $= \left[2(16) - \frac{64}{3}\right] = 32 - \frac{64}{3} = \frac{96}{3} - \frac{64}{3} = \frac{32}{3}$

So, the area under the curve is $\frac{32}{3}$ square units.

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Part c: $y = \sqrt{x}$

We need to find the area under the curve $y = \sqrt{x}$ from $x = 0$ to $x = 9$.

1. Set up the integral: $\text{Area} = \int_{0}^{9} \sqrt{x} \, dx$

2. Compute the integral: $\int \sqrt{x} \, dx = \int x^{1/2} \, dx = \frac{2}{3}x^{3/2}$

3. Evaluate from $0$ to $9$: [ \text{Area} = \left[\frac{2}{3}(9)^{3/2}\right] - \left[\frac{2}{3}(0)^{3/2}\right] ] $= \frac{2}{3} \times 27 = 18$

So, the area under the curve is 18 square units.

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Part d: $y = x(2 - x)(x + 3)$

This is a cubic function. We need to find the area under the curve between the given bounds, from $A$ to $B$.

1. Simplify the equation (if necessary): We could expand the function to make integration easier, but it's often more straightforward to leave it in factored form.

2. Set up the integral: Assuming $A$ and $B$ are the x-intercepts: $\text{Area} = \int_{A}^{B} x(2 - x)(x + 3) \, dx$

3. Compute the integral (detailed expansion omitted for now): Expand and integrate, then evaluate between the roots $A$ and $B$.

Given the complexity, if you specify the points $A$ and $B$, I can calculate it fully.

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Questions for Further Exploration:

1. How do definite integrals differ from indefinite integrals in their use and interpretation?
2. What methods can be used to solve integrals when the antiderivative is difficult to find directly?
3. How can integrals be applied to real-world problems, such as calculating areas or volumes?
4. What are some techniques to approximate the area under a curve if the integral is too complex to solve analytically?
5. How does the Fundamental Theorem of Calculus link differentiation and integration?

Tip:

Always double-check your bounds and the function you're integrating, especially if the curve crosses the x-axis, as this might require breaking the integral into parts where the function is positive or negative.