Chapter 5: Q11P (page 247)

Question. $\int \int_{A} x d x d y$ , where A is the area between the parabolarole="math" localid="1658895091680" $y = x^{2}$ and the straight line 2x-y+8=0.

Short Answer

Expert verified

The required solution is 36.

Step by step solution

Definition of double integral

The double integral of f(x,y) over the area A in 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 area between the parabola $y = x^{2}$ and the straight line 2x - y + 8 = 0.

Intersecting points identification

From the figure, the intersecting points are:

$x_{1, 2} = \frac{2 \pm \sqrt{4 + 32}}{2} = (4, - 2)$

Calculation of the area under the curve

The integral evaluates to:

$\int \int_{A} x d x d y = \int_{x = - 2}^{4} [\int_{y = x^{2}}^{2 x + 8} d y] x d x = \int_{- 2}^{4} (2 x + 8 - x^{2}) d x = {(\frac{2}{3} x^{3} + 4 x^{2} - \frac{1}{4} x^{4})}_{- 2}^{4} = 36$

Therefore, the area under the given graph is 36.

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Question. $\int \int_{A} x d x d y$ , where A is the area between the parabolarole="math" localid="1658895091680" $y = x^{2}$ and the straight line 2x-y+8=0.

Short Answer

Step by step solution

Definition of double integral

Drawing the area bounded by the curve

Intersecting points identification

Calculation of the area under the curve

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Question. ∫∫Axdxdy, where A is the area between the parabolarole="math" localid="1658895091680" y=x2 and the straight line 2x-y+8=0.

Short Answer

Step by step solution

Definition of double integral

Drawing the area bounded by the curve

Intersecting points identification

Calculation of the area under the curve

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Nuclear Physics

Solid State Physics

Work Energy and Power

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Electric Charge Field and Potential

Engineering Physics

Study anywhere. Anytime. Across all devices.

Question. $\int \int_{A} x d x d y$ , where A is the area between the parabolarole="math" localid="1658895091680" $y = x^{2}$ and the straight line 2x-y+8=0.