Explanation Given: Equation of given line is y = m x + b And m c + b > 0 , m > 0 Concept used: Integration of a function can be defined by anti derivative of that function. For any function f ( x ) = a x + C , derivative of the function can be evaluated as f ′ ( x ) = a d d x ( x ) + d d x ( C ) = a ( 1 ) + 0 = a Therefore, antiderivative of a is given by ∫ a d x = a x + C Here, a is any constant and C is integrating constant. Formula used: Formula to evaluate the integral of the form ∫ a x n d x is shown below. ∫ a x n d x = a ∫ x n d x = a x n + 1 n + 1 + C n ≠ − 1 Formula to evaluate the integral of the form ∫ b d x is shown below. ∫ b d x = b ∫ d x = b x + C Linear property of inderfinite integral are as follows: ∫ [ a f ( x ) ± b g ( x ) ] d x = a ∫ f ( x ) d x ± b ∫ g ( x ) d x Area theorem: If f is a continuous function such that f ( x ) ≥ 0 for all x on the closed interval [ a , b ] then the area bounded by the curve y = f ( x ) , the x-axis and the vertical lines x = a , x = t , viewed as a function of t, and is an antiderivative of f ( t ) on [ a , b ] . Calculation: Consider the given equation of curve. y = m x + b The region under the curve and above the x -axis on [ c , d ] is shown below.

6th Edition

ISBN: 9781465208880

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

Publisher: Kendall Hunt Publishing

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Chapter 5.1, Problem 60PS

To determine

To find:The area under the given line over the interval [c,d] above the x - axis .

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Chapter 5.1, Problem 59PS

Chapter 5.2, Problem 1PS

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12. [-/1 Points] DETAILS MY NOTES SESSCALCET2 6.3.508.XP. ASK YOUR TEA Make a substitution to express the integrand as a rational function and then evaluate the integral. (Remember to use absolute values where appropriate. Use C for the constant of integration.) x + 16 dx X Need Help? Read It SUBMIT ANSWER 13. [-/1 Points] DETAILS MY NOTES SESSCALCET2 6.3.512.XP. ASK YOUR TEA Make a substitution to express the integrand as a rational function and then evaluate the integral. (Remember to use absolute values where appropriate. Use C for the constant of integration.) dx 8)(2x + 1) Need Help? Read It SUBMIT ANSWER 14. [-/1 Points] DETAILS MY NOTES SESSCALCET2 6.3.518.XP. Find the area of the region under the given curve from 1 to 5. y = x² +7 6x - x² Need Help? Read It ASK YOUR TEA

DETAILS MY NOTES SESSCALCET2 6.3.012. 6. [-/1 Points] Evaluate the integral. x-4 dx x² - 5x + 6 Need Help? Read It SUBMIT ANSWER 7. [-/1 Points] DETAILS MY NOTES SESSCALCET2 6.3.019. Evaluate the integral. (Remember to use absolute values where appropriate. Use C for the constant of integration.) x²+1 (x-6)(x-5)² dx Need Help? Read It SUBMIT ANSWER 8. [-/1 Points] DETAILS MY NOTES SESSCALCET2 6.3.021. Evaluate the integral. (Remember to use absolute values where appropriate. Use C for the constant of integration.) ✓ x² 4 +4 dx

Chapter 5 Solutions

Calculus

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The area under the given line over the interval [ c , d ] above the x - axis .

Solution Summary: The author explains how integration of a function can be defined by anti-derivative of that function.

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To find:The area under the given line over the interval [c,d] above the x - axis .

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Chapter 5 Solutions