https://file.aiquickdraw.com/mathbot/file/17297947903749l5oiiqp.jpg

Let's address each part of the question one by one:

### Part (a)
We are asked to calculate the volume of the solid bounded by the cone \( z^2 = x^2 + y^2 \) and the plane \( z = 2 \) using a triple integral.

#### Step 1: Convert to cylindrical coordinates
In cylindrical coordinates, we know:
- \( x = r \cos \theta \)
- \( y = r \sin \theta \)
- \( z = z \)
- The relation \( z^2 = x^2 + y^2 \) becomes \( z^2 = r^2 \), or \( z = r \).

So, the volume is calculated by integrating the function \( 1 \) over the volume enclosed by the cone and the plane. The limits for the cylindrical coordinates are:
- \( r \) ranges from \( 0 \) to \( 2 \) (since \( z = 2 \))
- \( \theta \) ranges from \( 0 \) to \( 2\pi \)
- \( z \) ranges from \( r \) (the cone) to \( 2 \) (the plane)

#### Step 2: Set up the integral
The volume integral in cylindrical coordinates is:
\[
V = \int_0^{2\pi} \int_0^2 \int_r^2 r \, dz \, dr \, d\theta
\]

#### Step 3: Compute the integral
First, integrate with respect to \( z \):
\[
\int_r^2 dz = 2 - r
\]
Now the volume becomes:
\[
V = \int_0^{2\pi} \int_0^2 (2 - r) r \, dr \, d\theta
\]
Now integrate with respect to \( r \):
\[
\int_0^2 (2r - r^2) \, dr = \left[ r^2 - \frac{r^3}{3} \right]_0^2 = 4 - \frac{8}{3} = \frac{12}{3} - \frac{8}{3} = \frac{4}{3}
\]
Finally, integrate with respect to \( \theta \):
\[
V = \int_0^{2\pi} \frac{4}{3} \, d\theta = \frac{4}{3} \cdot 2\pi = \frac{8\pi}{3}
\]

Thus, the volume of the solid is:
\[
V = \frac{8\pi}{3}
\]

---

### Part (b)
We are asked to evaluate the double integral:
\[
\int_0^1 \int_0^{y^2} x y \, dx \, dy
\]
by changing the order of integration.

#### Step 1: Identify the region of integration
The region is described by \( 0 \leq y \leq 1 \) and \( 0 \leq x \leq y^2 \). We need to reverse the order of integration. The limits for \( x \) now range from \( 0 \) to \( 1 \), and for a given \( x \), \( y \) ranges from \( \sqrt{x} \) to \( 1 \).

#### Step 2: Rewrite the integral
We change the order of integration:
\[
\int_0^1 \int_{\sqrt{x}}^1 x y \, dy \, dx
\]

#### Step 3: Compute the integral
First, integrate with respect to \( y \):
\[
\int_{\sqrt{x}}^1 y \, dy = \left[ \frac{y^2}{2} \right]_{\sqrt{x}}^1 = \frac{1}{2} - \frac{x}{2}
\]
Now, the integral becomes:
\[
\int_0^1 x \left( \frac{1}{2} - \frac{x}{2} \right) \, dx = \frac{1}{2} \int_0^1 x \, dx - \frac{1}{2} \int_0^1 x^2 \, dx
\]
Evaluate these integrals:
\[
\int_0^1 x \, dx = \frac{1}{2}, \quad \int_0^1 x^2 \, dx = \frac{1}{3}
\]
Thus, the expression becomes:
\[
\frac{1}{2} \left( \frac{1}{2} \right) - \frac{1}{2} \left( \frac{1}{3} \right) = \frac{1}{4} - \frac{1}{6} = \frac{3}{12} - \frac{2}{12} = \frac{1}{12}
\]

So the value of the integral is:
\[
\frac{1}{12}
\]

---

Would you like further clarification on any part of this process, or additional details?

### 5 Related Questions:
1. How do you convert Cartesian coordinates to cylindrical coordinates?
2. What are the typical applications of triple integrals in real-world problems?
3. How do you interpret a region of integration graphically to change the order of integration?
4. How can you use polar coordinates for integrating over circular regions?
5. What are the steps for calculating the volume of a solid using a triple integral?

### Tip:
When changing the order of integration, always sketch the region of integration first to properly understand the bounds for each variable.

Using triple integral obtain the volume of the solid bounded between the cone z^2 = x^2 + y^2 and the plane z = 2. Evaluate ∫[0,1] ∫[0,y^2] xy dy dx by changing the order of integration.

Learn how to compute the volume of the solid bounded by the cone z^2 = x^2 + y^2 and the plane z = 2 using triple integrals in cylindrical coordinates. This problem also involves changing the order of integration for a double integral. Suitable for undergraduate math students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation