# Left-Hand Sum Exercises

### Example 1

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

• Draw R and the 8 rectangles that result from using a Left-Hand Sum with 8 sub-intervals to approximate the area of R.
• Use the Left-Hand sum with 8 sub-intervals to approximate the area of R (you might want a calculator).

### 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 Left-Hand Sum with 2 sub-intervals to approximate the area of S. Draw S and the rectangles used in this Left-Hand Sum on the same graph.
• Use a Left-Hand Sum with 5 sub-intervals to approximate the area of S. Draw S and the rectangles used in this Left-Hand Sum on the same graph.
• Are the approximations 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.
• Use a Left-Hand Sum with 3 subintervals to approximate the area of W. Draw W and the rectangles used in this Left-Hand Sum on the same graph.
• Use a Left-Hand Sum with 6 subintervals to approximate the area of W. Draw W and the rectangles used in this Left-Hand Sum on the same graph.
• Are your approximations bigger or smaller than the actual area of W?

### Example 4

• The table below shows some values of an increasing function.

• Use a left-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 left-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 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 left-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 left-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 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 f (x) = x2 + 6x + 9. Use a left-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 7

• Let f(x) = -x2 + 2x + 8. Use a left-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 8

Let g be a function with values given by the table below.

Use a left-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 9

Let h be a function with values given by the table below. Use a left-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 10

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

### Example 11

Use a left-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].

### Example 12

Use a left-hand sum with the sub-intervals indicated by the data in the table to estimate the area between the graph of g and the x-axis on the interval [0, 5π].

### Example 13

Use a left-hand sum to estimate the area between the graph of h and the x-axis on the interval [2, 7].