Chapter 14: Q. 21 (page 1119)

Find the area of S is the portion of the plane with equation y−z = $\frac{π}{2}$
that lies above the rectangle determined by 0 ≤ x ≤ 4 and 3 ≤ y ≤ 6.

Short Answer

Expert verified

$Hence the area of triangle is = 12 \sqrt{2} .$

Step by step solution

Step 1. Given

Equation of plane is y−z = $\frac{π}{2}$

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

$y - z = \frac{π}{2} t h e n, z = y - \frac{π}{2} n o w, f i r s t f i n d \begin{aligned} \frac{\partial z}{\partial x} = \frac{\partial}{\partial x} (y - \frac{π}{2}) \\ = 0 \end{aligned} a n d \begin{aligned} \frac{\partial z}{\partial y} = \frac{\partial}{\partial y} (y - \frac{π}{2}) \\ = 1 \end{aligned}$

Step 4. Finding area

$T h e a r e a o f s u r f a c e i s \begin{aligned} \int_{S} d S = \iint_{D} \sqrt{{(\frac{\partial z}{\partial x})}^{2} + {(\frac{\partial z}{\partial y})}^{2} + 1} d A \\ = \int_{3}^{6} \int_{0}^{4} \sqrt{0^{2} + 1^{2} + 1} d x d y \\ = \int_{3}^{6} \int_{0}^{4} \sqrt{2} d x d y \\ = \sqrt{2} \int_{3}^{6} \int_{0}^{4} d x d y \\ = \sqrt{2} \int_{3}^{6} [\int_{0}^{4} d x] d y \\ = \sqrt{2} \int_{3}^{6} [x]_{0}^{4} d y \\ = \sqrt{2} \int_{3}^{6} [4 - 0] d y \\ = \sqrt{2} \int_{3}^{6} 4 d y \\ = 4 \sqrt{2} \int_{3}^{6} d y \\ = 4 \sqrt{2} [y]_{3}^{6} \\ = 4 \sqrt{2} [6 - 3] \\ = 4 \sqrt{2} [3] \\ = 12 \sqrt{2} . \\ Hence the area of triangle is = 12 \sqrt{2} . \end{aligned}$

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Find the area of S is the portion of the plane with equation y−z = $\frac{π}{2}$
that lies above the rectangle determined by 0 ≤ x ≤ 4 and 3 ≤ y ≤ 6.

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. Finding area

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Find the area of S is the portion of the plane with equation y−z =π2that lies above the rectangle determined by 0 ≤ x ≤ 4 and 3 ≤ y ≤ 6.

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. Finding area

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Logic and Functions

Pure Maths

Applied Mathematics

Theoretical and Mathematical Physics

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

Find the area of S is the portion of the plane with equation y−z = $\frac{π}{2}$
that lies above the rectangle determined by 0 ≤ x ≤ 4 and 3 ≤ y ≤ 6.