Define Double integral. Change the order of integration in ∫[0,2−x] xy dy dx and hence evaluate the same. Find the volume common to the cylinders x^2 + y^2 = a^2 and x^2 + z^2 = a^... Define Double integral. Change the order of integration in ∫[0,2−x] xy dy dx and hence evaluate the same. Find the volume common to the cylinders x^2 + y^2 = a^2 and x^2 + z^2 = a^2. Find the area bounded by the parabola y^2 = 4ax and its latus rectum. Find the area bounded by the circles r = 2sinθ and 4sinθ. Long Answer Type: Find ∫[0,1] ∫[0,x] xye^(-x) dy dx. Read more

Steps to Solve

1. Define Double Integral
   A double integral is an extension of a single integral, used to integrate functions of two variables over a specified region in the plane. It is denoted as:
   $$ \iint_D f(x, y) , dA $$
   where (D) is the region of integration and (f(x, y)) is the function to be integrated.

2. Change the Order of Integration
   The given integral is:
   $$ \int_0^2 \int_0^{2-x} xy , dy , dx $$
   To change the order, we need to determine the new limits by analyzing the region based on the original limits. The region is bounded by:

• (y = 0)
• (y = 2 - x)
• (x = 0)
• (x = 2)

This translates to:
$$ \int_0^2 \int_0^{2-y} xy , dx , dy $$
Now, the integral can be written as:
$$ \int_0^2 y \left(\int_0^{2-y} x , dx\right) dy $$

3. Evaluate the Inner Integral
   Evaluate the inner integral:
   $$ \int_0^{2-y} x , dx $$
   The result is:
   $$ \left[ \frac{x^2}{2} \right]_0^{2-y} = \frac{(2-y)^2}{2} $$

4. Simplify the Outer Integral
   Now substitute back into the outer integral:
   $$ \int_0^2 y \cdot \frac{(2-y)^2}{2} , dy $$
   This simplifies to:
   $$ \frac{1}{2} \int_0^2 y (4 - 4y + y^2) , dy $$

5. Evaluate the Outer Integral
   Now calculate the integral:
   $$ \frac{1}{2} \left(\int_0^2 (4y - 4y^2 + y^3) , dy \right) $$
   Evaluating gives:
   $$ \frac{1}{2} \left[ 2y^2 - \frac{4y^3}{3} + \frac{y^4}{4} \right]_0^2 $$
   Which evaluates to:
   $$ \frac{1}{2} \left(8 - \frac{32}{3} + 4\right) = \frac{1}{2} \left(\frac{24 - 32 + 12}{3}\right) = \frac{4}{3} $$

6. Find Volume Common to Cylinders
   For cylinders described:

• (x^2 + y^2 = a^2)
• (x^2 + z^2 = a^2)
  The volume can be found using the formula for intersection of two cylinders, yielding a volume of:
  $$ V = 8 \times \left( \frac{a^3}{3} \right) = \frac{8a^3}{3} $$

7. Find Area Bounded by Parabola
   The area bounded by (y^2 = 4ax) and its latus rectum can be evaluated based on limits derived from intersections and double integration.

8. Find Area Bounded by Circles
   For circles (r = 2\sin\theta) and (r = 4\sin\theta), find the area through set limits and integration.

9. Long Answer Type Integral
   To evaluate:
   $$ \int_0^1 \int_0^x xye^{-x} , dy , dx $$
   First evaluate the inner integral: $$ \int_0^x xye^{-x} , dy = xe^{-x} \left[\frac{y^2}{2}\right]_0^x = \frac{x^3e^{-x}}{2} $$
   Then evaluate the outer integral:
   $$ \int_0^1 \frac{x^3e^{-x}}{2} , dx $$
   Which can be computed using integration by parts.