Math Problem Statement

Sketch the region of integration, reverse the order of integration, and evaluate the integral: ∫∫ (from y = √x to y = √3) (from x = 0 to x = √3) 3y dy dx.

Solution

The problem asks us to:

  1. Sketch the region of integration.
  2. Reverse the order of integration.
  3. Evaluate the integral.

Given the double integral:

03x33ydydx\int_0^{\sqrt{3}} \int_{\sqrt{x}}^{\sqrt{3}} 3y \, dy \, dx

Let's go through these steps:

1. Sketching the Region of Integration

The integral has the outer limits for xx from 00 to 3\sqrt{3}, and the inner limits for yy from x\sqrt{x} to 3\sqrt{3}.

The bounds describe the region:

  • xx ranges from 00 to 3\sqrt{3}.
  • yy ranges from x\sqrt{x} to 3\sqrt{3}.

The curve y=xy = \sqrt{x} defines the lower boundary for yy, and the line y=3y = \sqrt{3} is the upper boundary.

2. Reversing the Order of Integration

To reverse the order of integration, we need to describe xx in terms of yy.

  • From y=xy = \sqrt{x}, we get x=y2x = y^2.
  • xx ranges from 00 to 3\sqrt{3}, and yy ranges from 00 to 3\sqrt{3}.

The region can be rewritten with yy ranging from 00 to 3\sqrt{3} and xx ranging from 00 to y2y^2. Thus, the reversed integral becomes:

030y23ydxdy\int_0^{\sqrt{3}} \int_0^{y^2} 3y \, dx \, dy

3. Evaluating the Integral

Now, let's evaluate the reversed integral:

030y23ydxdy=033y[x]0y2dy=033yy2dy=033y3dy\int_0^{\sqrt{3}} \int_0^{y^2} 3y \, dx \, dy = \int_0^{\sqrt{3}} 3y \left[x\right]_0^{y^2} \, dy = \int_0^{\sqrt{3}} 3y \cdot y^2 \, dy = \int_0^{\sqrt{3}} 3y^3 \, dy

=303y3dy=3[y44]03=3(3)44=394=274= 3 \int_0^{\sqrt{3}} y^3 \, dy = 3 \left[\frac{y^4}{4}\right]_0^{\sqrt{3}} = 3 \cdot \frac{(\sqrt{3})^4}{4} = 3 \cdot \frac{9}{4} = \frac{27}{4}

Therefore, the value of the integral is:

274\boxed{\frac{27}{4}}

Would you like more details or have any questions?

Here are 5 related questions to expand on this topic:

  1. How can you determine the region of integration when given different integral limits?
  2. What are some common methods to reverse the order of integration?
  3. How do changes in bounds affect the area of the integration region?
  4. What are some applications of double integrals in real-world problems?
  5. How can changing variables (like using polar coordinates) simplify certain double integrals?

Tip: Always sketch the region of integration before reversing the order; it helps in setting up the correct bounds for the new order of integration.

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Math Problem Analysis

Mathematical Concepts

Double Integrals
Reversing Order of Integration
Integration Techniques
Region of Integration

Formulas

Double Integral: ∫∫ f(x, y) dA
Reversing Order of Integration: Changing limits based on region boundaries

Theorems

Fubini's Theorem for Double Integrals

Suitable Grade Level

College Calculus (1st Year)