Chapter 14: Q. 22 (page 1119)

Find the area of S is the portion of the plane with equation x = y + z that lies above the region in the xy-plane that is bounded by y = x, y = 5, y = 10, and the y-axis.

Short Answer

Expert verified

$Hence the area of triangle is = \frac{75 \sqrt{3}}{2}$

Step by step solution

Step 1. Given

Equation of plane x=y+ z

Step 2. Formula for finding the area of surface

$If a surface S is given by z = f (x, y) for f (x, y) \in D \subset R^{2} Then the surface area of smooth surface is \begin{aligned} \int_{S} d S = \int_{S} \sqrt{{(\frac{\partial z}{\partial x})}^{2} + {(\frac{\partial z}{\partial y})}^{2} + 1 d A} \\ = \iint_{D} \sqrt{{(\frac{\partial z}{\partial x})}^{2} + {(\frac{\partial z}{\partial y})}^{2} + 1 d A \dots \dots (1} \end{aligned}$

Step 3. Finding the partial derivative

$x = y + z t h e n, z = x - y n o w, f i r s t f i n d \begin{aligned} \frac{\partial z}{\partial x} = \frac{\partial}{\partial x} (x - y) \\ = 1 \end{aligned} a n d \begin{aligned} \frac{\partial z}{\partial y} = \frac{\partial}{\partial y} (x - y) \\ = - 1 \end{aligned}$

Step 4. Graph

viewed it as a x-simple, then the region of integration will be

$D = \{(x, y) | 0 \leq x \leq y, 5 \leq y \leq 10\}$

Step 5. Finding the area

$T h e a r e a o f s u r f a c e i s \begin{aligned} \int_{S} d S = \iint_{D} \sqrt{{(\frac{\partial z}{\partial x})}^{2} + {(\frac{\partial z}{\partial y})}^{2} + 1} d A \\ = \int_{5}^{10} \int_{0}^{y} \sqrt{1^{2} + - 1^{2} + 1} d x d y \\ = \int_{5}^{10} \int_{0}^{y} \sqrt{3} d x d y \\ = \sqrt{3} \int_{5}^{10} \int_{0}^{y} d x d y \\ = \sqrt{3} \int_{5}^{10} \int_{0}^{y} d x d y \\ = \sqrt{3} \int_{5}^{10} [x]_{0}^{y} d y \\ = \sqrt{3} \int_{5}^{10} [y - 0] d y \\ = \sqrt{3} \int_{5}^{10} y d y \\ = \sqrt{3} \int_{5}^{10} y d y \\ = \sqrt{2} [\frac{y^{2}}{2}]_{5}^{10} \\ = \sqrt{2} [\frac{10^{2}}{2} - \frac{5^{2}}{2}] \\ = \frac{75 \sqrt{3}}{2} \\ Hence the area of triangle is = \frac{75 \sqrt{3}}{2} \end{aligned}$

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Find the area of S is the portion of the plane with equation x = y + z that lies above the region in the xy-plane that is bounded by y = x, y = 5, y = 10, and the y-axis.

Short Answer

Step by step solution

Step 1. Given

Step 2. Formula for finding the area of surface

Step 3. Finding the partial derivative

Step 4. Graph

Step 5. Finding the area

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