Chapter 5: Q13P (page 247)

$\int \int 2 x y d x d y$ over the triangle with vertices role="math" localid="1658828431182" $(0, 0) (, 2, 1), (3, 0)$ .

Short Answer

Expert verified

The required solution is $\frac{7}{4}$ .

Step by step solution

Definition of double integral

The double integral of $f (x, y)$ over the area Ain the $(x, y)$ plane as the limit of this sum, and we write it as $\int \int_{A} f (x, y) d x d y$ .

Drawing the area bounded by the curve

The triangle with vertices (0,0),(2,1) and (3,0).

Integration over the bounded curve

Now the total integral region can be divided into two parts and then adding them can be found in the final area.

$\int \int_{A} y d y d x = I = I_{1} + I_{2}$

The area of the left triangle

Calculation of the value, $I_{1}$ :

$I_{1} = \int_{X = 0}^{2} [\int_{Y = 0}^{X / 2} 2 ydy] xdx = \int_{0}^{2} ({y^{2}}_{0}^{x / 2}) xdx = \frac{1}{4} \int_{0}^{2} x^{3} dx = {\frac{1}{16} x^{4}}_{0}^{2} = 1$

The area of the right side triangle

Calculation of the value, $I_{2}$ :

$I_{2} = \int_{X = 2}^{3} [\int_{Y = 0}^{3 - X} 2 ydy] xdx = \int_{2}^{3} {(3 - x)}^{2} xdx = {(\frac{9}{2} x^{2} - 2 x^{3} + \frac{1}{4} x^{4})}_{2}^{3} = \frac{3}{4}$

Total area bounded by the triangle

Hence:

$I = I_{1} + I_{2} = 1 + \frac{3}{4} = \frac{7}{4}$

Therefore, the value is $\frac{7}{4}$ .

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$\int \int 2 x y d x d y$ over the triangle with vertices role="math" localid="1658828431182" $(0, 0) (, 2, 1), (3, 0)$ .

Short Answer

Step by step solution

Definition of double integral

Drawing the area bounded by the curve

Integration over the bounded curve

The area of the left triangle

The area of the right side triangle

Total area bounded by the triangle

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∫∫2xydxdyover the triangle with vertices role="math" localid="1658828431182" (0,0)(,2,1),(3,0).

Short Answer

Step by step solution

Definition of double integral

Drawing the area bounded by the curve

Integration over the bounded curve

The area of the left triangle

The area of the right side triangle

Total area bounded by the triangle

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Medical Physics

Waves Physics

Turning Points in Physics

Fields in Physics

Conservation of Energy and Momentum

Particle Model of Matter

Study anywhere. Anytime. Across all devices.

$\int \int 2 x y d x d y$ over the triangle with vertices role="math" localid="1658828431182" $(0, 0) (, 2, 1), (3, 0)$ .