Explanation Given: Given planes are x = 0 y = 0 z = 0 x a + y b + z c = 1 Concept used: Volume of tetrahedron = ∫ ∫ ∫ d z d y d x To evaluate a triple integral, first z variable is integrated and then y variable is integrated and at last x variable is integrated. Calculation: x a + y b + z c = 1..... ( i ) ⇒ z c = 1 − ( x a + y b ) ⇒ z c = 1 − x a − y b ⇒ z = c ( 1 − x a − y b ) Limits of z : 0 to c ( 1 − x a − y b ) To, get limits of y put z = 0 in equation (i), x a + y b = 1 ⇒ y b = 1 − x a ⇒ y = b ( 1 − x a ) Limits of y : 0 to b ( 1 − x a ) To get limits of x put y and z equal to zero in equation (i), x a = 1 ⇒ x = a Limits of x : 0 to a Volume of tetrahedron is: ⇒ ∫ ∫ ∫ d z d y d x ⇒ ∫ 0 a ∫ 0 b ( 1 − x a ) ∫ 0 c ( 1 − x a − y b ) d z d y d x ⇒ ∫ 0 a ∫ 0 b ( 1 − x a ) ( ∫ 0 c ( 1 − x a − y b ) d z ) d y d x ⇒ ∫ 0 a ∫ 0 b ( 1 − x a ) ( ∫ 0 c ( 1 − x a − y b ) d z ) d y d x ⇒ ∫ 0 a ∫ 0 b ( 1 − x a ) ( [ z ] 0 c ( 1 − x a − y b ) ) d y d x ⇒ ∫ 0 a ∫ 0 b ( 1 − x a ) ( c ( 1 − x a − y b ) − 0 ) d y d x ⇒ ∫ 0 a ∫ 0 b ( 1 − x a ) ( c ( 1 − x a − y b ) ) d y d x ⇒ ∫ 0 a ∫ 0 b ( 1 − x a ) ( c − c x a − c y b ) d y d x ⇒ ∫ 0 a ∫ 0 b ( 1 − x a ) c − c x a − c y b d y d x ⇒ ∫ 0 a ( ∫ 0 b ( 1 − x a ) c × 1 − c x a × 1 − c b × y d y ) d x ⇒ ∫ 0 a ( [ c y − c x a × y − c b × y 2 2 ] 0 b ( 1 − x a ) ) d x ⇒ ∫ 0 a ( c [ { b ( 1 − x a ) } − x a × { b ( 1 − x a ) } − 1 b × { b ( 1 − x a ) } 2 2 ] ) d x ⇒ c ∫ 0 a (

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Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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Chapter 12, Problem 26PE

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To find the volume of tetrahedron that is bounded by the coordinate planes and the plane xa+yb+zc=1

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(b) Find the (instantaneous) rate of change of y at x = 5. In the previous part, we found the average rate of change for several intervals of decreasing size starting at x = 5. The instantaneous rate of change of fat x = 5 is the limit of the average rate of change over the interval [x, x + h] as h approaches 0. This is given by the derivative in the following limit. lim h→0 - f(x + h) − f(x) h The first step to find this limit is to compute f(x + h). Recall that this means replacing the input variable x with the expression x + h in the rule defining f. f(x + h) = (x + h)² - 5(x+ h) = 2xh+h2_ x² + 2xh + h² 5✔ - 5 )x - 5h Step 4 - The second step for finding the derivative of fat x is to find the difference f(x + h) − f(x). - f(x + h) f(x) = = (x² x² + 2xh + h² - ])- = 2x + h² - 5h ])x-5h) - (x² - 5x) = ]) (2x + h - 5) Macbook Pro

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To find the volume of tetrahedron that is bounded by the coordinate planes and the plane x a + y b + z c = 1

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To find the volume of tetrahedron that is bounded by the coordinate planes and the plane xa+yb+zc=1

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