Chapter 5: Q15P (page 247)

$\iint d x d y$ over the area bounded by $y = Inx, y = e + 1 - x$ and the x axis.

Short Answer

Expert verified

The required solution is $\frac{3}{2}$ .

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 $\iint_{A} f (x, y) d x d y$

Drawing the area bounded by the curve

The area is bounded by the graph $y = I n x, y = e + 1 - x$ and x-axis.

Intersecting points identification

From the figure, the two lines $y = I n x, y = e + 1 - x$ intersect at $x = e$ .

The integral area splits into two parts. First, have to calculate individual areas then adding both of them gives the total area bounded by the region.

Calculation of the left area bounded by the curve

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

$I_{1} = \int_{x = 1}^{e} [\int_{y = 0}^{Inx} dy] dx = \int_{1}^{e} Inxdx = x {z (Inx - 1)}_{1}^{e} = 1$

Step 5. Calculation of the right side area bounded by the curve

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

$I_{2} = \int_{x = e}^{e + 1} [\int_{y = 0}^{e + 1 - x} d y] d x = \int_{e}^{e + 1} (e + 1 - x) d x = {((e + 1) x - \frac{1}{2} x^{2})}_{e}^{e + 1} = \frac{1}{2}$

Total area bounded by the curve

Hence:

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

Therefore, the value is $\frac{3}{2}$ .

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$\iint d x d y$ over the area bounded by $y = Inx, y = e + 1 - x$ and the x axis.

Short Answer

Step by step solution

Definition of double integral

Drawing the area bounded by the curve

Intersecting points identification

Calculation of the left area bounded by the curve

Step 5. Calculation of the right side area bounded by the curve

Total area bounded by the curve

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∬dxdyover the area bounded byy=Inx,y=e+1-x and the x axis.

Short Answer

Step by step solution

Definition of double integral

Drawing the area bounded by the curve

Intersecting points identification

Calculation of the left area bounded by the curve

Step 5. Calculation of the right side area bounded by the curve

Total area bounded by the curve

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Circular Motion and Gravitation

Modern Physics

Electric Charge Field and Potential

Solid State Physics

Physical Quantities And Units

Mechanics and Materials

Study anywhere. Anytime. Across all devices.

$\iint d x d y$ over the area bounded by $y = Inx, y = e + 1 - x$ and the x axis.