Explanation Given: 0 < a < b are real numbers. Concept Used: The definite integral ∫ a b f ( x ) d x represents the area between the graph of f ( x ) and x -axis on the interval [ a , b ] . The area that is bounded by the graph of f ( x ) above the x -axis is positive. The area that is bounded by the graph of f ( x ) below the x -axis is negative. Area of trapezoid = 1 2 ( Sum of parallel sides ) × height Calculation: In order to prove that the definite integral ∫ a b x d x = 1 2 ( b 2 − a 2 ) , first plot the graph of the function f ( x ) = x on the interval [ a , b ] . And then break the region bounded by the graph and x -axis using some geometric shapes like triangles, rectangles or circles and find their area. Use the concept that the definite integral ∫ a b f ( x ) d x represents the area between the graph of f ( x ) and x -axis on the interval [ a , b ]

14th Edition

ISBN: 9781464151088

Author: Associate Professor Laura Taalman, Peter Kohn

Publisher: W.H. Freeman & Company

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Chapter 4.3, Problem 59P

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To prove that ∫abx dx=12(b2−a2) for any real numbers 0<a<b , using geometry.

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Chapter 4.3, Problem 58P

Chapter 4.3, Problem 60P

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A graph of the function f is given below: Study the graph of ƒ at the value given below. Select each of the following that applies for the value a = 1 Of is defined at a. If is not defined at x = a. Of is continuous at x = a. If is discontinuous at x = a. Of is smooth at x = a. Of is not smooth at = a. If has a horizontal tangent line at = a. f has a vertical tangent line at x = a. Of has a oblique/slanted tangent line at x = a. If has no tangent line at x = a. f(a + h) - f(a) lim is finite. h→0 h f(a + h) - f(a) lim h->0+ and lim h h->0- f(a + h) - f(a) h are infinite. lim does not exist. h→0 f(a+h) - f(a) h f'(a) is defined. f'(a) is undefined. If is differentiable at x = a. If is not differentiable at x = a.

The graph below is the function f(z) 4 3 -2 -1 -1 1 2 3 -3 Consider the function f whose graph is given above. (A) Find the following. If a function value is undefined, enter "undefined". If a limit does not exist, enter "DNE". If a limit can be represented by -∞o or ∞o, then do so. lim f(z) +3 lim f(z) 1-1 lim f(z) f(1) = 2 = -4 = undefined lim f(z) 1 2-1 lim f(z): 2-1+ lim f(x) 2+1 -00 = -2 = DNE f(-1) = -2 lim f(z) = -2 1-4 lim f(z) 2-4° 00 f'(0) f'(2) = = (B) List the value(s) of x for which f(x) is discontinuous. Then list the value(s) of x for which f(x) is left- continuous or right-continuous. Enter your answer as a comma-separated list, if needed (eg. -2, 3, 5). If there are none, enter "none". Discontinuous at z = Left-continuous at x = Invalid use of a comma.syntax incomplete. Right-continuous at z = Invalid use of a comma.syntax incomplete. (C) List the value(s) of x for which f(x) is non-differentiable. Enter your answer as a comma-separated list, if needed (eg. -2, 3, 5).…

A graph of the function f is given below: Study the graph of f at the value given below. Select each of the following that applies for the value a = -4. f is defined at = a. f is not defined at 2 = a. If is continuous at x = a. Of is discontinuous at x = a. Of is smooth at x = a. f is not smooth at x = a. If has a horizontal tangent line at x = a. f has a vertical tangent line at x = a. Of has a oblique/slanted tangent line at x = a. Of has no tangent line at x = a. f(a + h) − f(a) h lim is finite. h→0 f(a + h) - f(a) lim is infinite. h→0 h f(a + h) - f(a) lim does not exist. h→0 h f'(a) is defined. f'(a) is undefined. If is differentiable at x = a. If is not differentiable at x = a.

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To prove that ∫ a b x d x = 1 2 ( b 2 − a 2 ) for any real numbers 0 < a < b , using geometry.

Solution Summary: The author explains that the definite integral displaystyle 'int' is the area between the graph and x -axis.

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To prove that ∫abx dx=12(b2−a2) for any real numbers 0<a<b , using geometry.

Want to see the full answer?

Chapter 4 Solutions

To prove that ∫ a b x d x = 1 2 ( b 2 − a 2 ) for any real numbers 0 &lt; a &lt; b , using geometry.

Solution Summary: The author explains that the definite integral displaystyle 'int' is the area between the graph and x -axis.

Videos

To prove that ∫abx dx=12(b2−a2) for any real numbers 0<a<b , using geometry.

Want to see the full answer?

Chapter 4 Solutions

To prove that ∫ a b x d x = 1 2 ( b 2 − a 2 ) for any real numbers 0 < a < b , using geometry.