Chapter 5: Q12P (page 247)

$\int \int ydxdy$ over the triangle with vertices $(- 1, 0), (0, 2), and (2, 0)$

Short Answer

Expert verified

The required solution is S

Step by step solution

Definition of double integral

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

Drawing the area bounded by the curve

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

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.

$\int \int_{A} ydydx = l = l_{1} + l_{2}$

The area of the left triangle

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

role="math" localid="1658895301367" $l_{1} = \int_{x = 1}^{0} [\int_{y = 1}^{2 + 2 x} ydy] dx = \int_{- 1}^{0} \frac{1}{2} {(2 + 2)}^{2} dx = \int_{0}^{1} {(1 + x)}^{2} d (1 + x) = \frac{2}{3} {(1 + x)}^{3}_{0}^{1} = \frac{2}{3}$

The area of the right side triangle

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

$l_{2} = \int_{x = 0}^{2} [\int_{y = 0}^{2 - x} ydy] dx = \int_{0}^{2} \frac{1}{2} {(2 + 2)}^{2} d (2 - x) = \frac{1}{6} {(2 + x)}^{3}_{0}^{2} = \frac{4}{3}$

Total area bounded by the triangle

Hence:

$l = l_{1} + l_{2} = 2$

Therefore, the value is 2.

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$\int \int ydxdy$ over the triangle with vertices $(- 1, 0), (0, 2), and (2, 0)$

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.

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∫∫ydxdyover the triangle with vertices (-1,0),(0,2),and(2,0)

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

Torque and Rotational Motion

Magnetism

Physics of Motion

Radiation

Quantum Physics

Classical Mechanics

Study anywhere. Anytime. Across all devices.

$\int \int ydxdy$ over the triangle with vertices $(- 1, 0), (0, 2), and (2, 0)$