https://r2.erweima.ai/mathbot/image/1725489140305a2udf2sx.jpg

Let's go through the questions on the image one by one:

### Question 1: 
**Use a definite integral to calculate the area of the following region. (35%)**

The region shown seems to be a semi-circle with radius 4 units. To calculate the area of this region using a definite integral, we can represent the upper half of the circle with the equation:

\[
x^2 + y^2 = 16 \quad \text{or} \quad y = \sqrt{16 - x^2}
\]

We will calculate the area by integrating the function over the x-axis from \(-4\) to \(4\):

\[
A = 2 \int_{0}^{4} \sqrt{16 - x^2} \, dx
\]

This is a standard integral for calculating the area of a semi-circle.

---

### Question 2: 
**Use a definite integral to find the area of the region. (25%)**

This question shows a cone-like shape. To find the area, you typically use a definite integral based on the boundaries of the function provided. The sketch suggests using a function of revolution or slicing.

Assuming the function represents a solid of revolution, we might apply methods like integrating a function related to the shape’s cross-section. Depending on the form of the function, it will likely involve:

\[
A = \int_{a}^{b} f(x) \, dx
\]

---

### Question 3:
**Calculate the volume of the solid region bounded by the paraboloid \(z = 36 - x^2 - y^2\) (40%)**

The volume of a solid defined by a paraboloid can be calculated using a double integral in polar coordinates. The equation \(z = 36 - x^2 - y^2\) is a paraboloid that opens downward. In polar coordinates, we transform the equation into:

\[
z = 36 - r^2
\]

The volume integral becomes:

\[
V = \int_0^{2\pi} \int_0^6 (36 - r^2) r \, dr \, d\theta
\]

Here, the limits of integration for \(r\) are from 0 to 6 (since at \(r = 6\), \(z = 0\)), and for \(\theta\), from 0 to \(2\pi\).

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Let me know if you'd like any further calculations on these problems.

### Questions:
1. Would you like to solve the integral for the semi-circle in Question 1 step-by-step?
2. Should we assume revolution in Question 2 for exact calculations?
3. Do you need the detailed solution for the volume of the paraboloid in Question 3?
4. Would you like to explore the use of polar coordinates for calculating the volume in Question 3?
5. Would you like to verify the sketch's details to ensure the functions are correct?

### Tip:
When solving problems involving regions and solids, always ensure that the integral bounds match the limits of the geometric shape being described.

Resolve the following problems using definite integrals, showing all calculations.

Explore the use of definite integrals to calculate the area of a semi-circle, the area of a region using integrals, and the volume of a paraboloid. This problem set covers essential calculus concepts such as integration, area calculation, and volume determination using polar coordinates, suitable for advanced high school or college students.

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