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Definite Integrals

Definite Integrals

Example 1

Let R be the region between the graph y = f ( x ) = x2 + 1 and the x-axis on the interval [0,4].

  • Draw R and the 8 rectangles that result from using a right-hand sum with 8 sub-intervals to approximate the area of R.
  • Use a Right-Hand Sum with 8 sub-intervals to approximate the area of R.
  • Is your approximation an under-estimate or an over-estimate to the actual area of R?

Example 2

Let S be the area between the graph of y = f ( x ) = 2x and the x-axis on the interval [1,6].

  • Draw S. 
  • Use a Right-Hand Sum with 2 subintervals to approximate the area of S. Draw S and the rectangles used in this Right-Hand Sum on the same graph.
  • Use a Right-Hand Sum with 5 subintervals to approximate the area of S. Draw S and the rectangles used in this Right-Hand Sum on the same graph. 
  • Are your approximations in parts (b) and (c) bigger or smaller than the actual area of S?

Example 3

Let W be the area between the graph of  and the x-axis on the interval [1,4]. 

  • Draw W.\item Use a Right-Hand Sum with 3 subintervals to approximate the area of W. Draw W and the rectangles used in this Right-Hand Sum on the same graph.
  • Use a Right-Hand Sum with 6 subintervals to approximate the area of W. Draw W and the rectangles used in this Right-Hand Sum on the same graph. 
  • Are your approximations in parts (b) and (c) bigger or smaller than the actual area of W?

Example 4

The table below shows some values of the increasing function f ( x ).

  • Use a right-hand sum with one sub-interval to estimate the area between the graph of f and the x-axis on the interval [2,8].
  • Use a right-hand sum with three sub-intervals to estimate the area between the graph of f and the x-axis on the interval [2,8].
  • Are your answers in (a) and (b) over- or under-estimates of the actual area between the graph of f and the x-axis on the interval [2,8]?

Example 5

Some values of the decreasing function g are given in the table below:

  • Use a right-hand sum with 3 sub-intervals to estimate the area between the graph of g and the x-axis on the interval [-1,2].
  • Use a right-hand sum with 2 sub-intervals to estimate the area between the graph of g and the x-axis on the interval [-1,2].
  • Are your answers in (a) and (b) over- or under-estimates for the actual area between the graph of g and the x-axis on the interval [-1,2]?

Example 6

  • Let W be the region between the graph of f and the x-axis on the interval [-20,20].

Use a right-hand sum with 4 sub-intervals to estimate the area of W.

Example 7

  • Let Z be the region between the graph of g and the x-axis on the interval [-4,0].

  • Use a right-hand sum with 2 sub-intervals to estimate the area of Z.
  • Use a right-hand sum with 4 sub-intervals to estimate the area of Z.
  • Are your answers in (a) and (b) over- or under- estimates for the area of Z?

Example 8

Let f ( x ) = x2 + 6x + 9. Use a right-hand sum with 6 sub-intervals to estimate the area between the graph of f and the x-axis on the interval [-6,-3].

Example 9

Let f ( x ) = -x2 + 2x + 8. Use a right-hand sum with 8 sub-intervals to estimate the area between the graph of f and the x-axis on the interval [0,4].

Example 10

Let g be a function with values given by the table below. Use a right-hand sum with 3 sub-intervals to estimate the area between the graph of g and the x-axis on the interval [0,12]. 

Example 11

Let h be a function with values given by the table below. Use a right-hand sum with 9 sub-intervals to estimate the area between the graph of h and the x-axis on the interval [-9,9]. 

Example 12

The function f ( x ) on the interval [0,30] is graphed below. Use a right-hand sum with 3 sub-intervals to estimate the area between the graph of f and the x-axis on this interval.

Example 13

Use a right-hand sum with the sub-intervals indicated by the data in the table to estimate the area between the graph of f and the x-axis on the interval [-10,1].

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