Worksheet: Numerical Integration: Riemann Sums

Q1:

The table shows the values of a function obtained from an experiment. Estimate using three equal subintervals with left endpoints.

	5	7	9	11	13	15	17
				0.4	1.8	2.5	3.1

Q2:

Using the midpoint rule with , round to four decimal places.

Q3:

The table shows the values of a function obtained from an experiment. Estimate using three equal subintervals with midpoints.

	1	2	3	4	5	6	7
					0.9	2.1	3.1

Q4:

Estimate using the midpoint rule with , giving your answer to four decimal places.

Q5:

Estimate using the midpoint rule with , giving your answer to four decimal places.

Q6:

Using the Midpoint Rule with , give an estimate of . Give your answer to four decimal places.

Q7:

Approximate the integral using a Riemann sum with right endpoints. Take the number of rectangles to be 8.

A
B
C
D
E

Q8:

The table gives the values of a function obtained from an experiment. Use them to estimate using three equal subintervals with right endpoints.

	3	7	11	15	19	23	27
				0.8	2.3	3.4	4.8

Q9:

Calculate the midpoint rule estimate of with subintervals. Is the result an overestimate or underestimate of the actual value?

A16, an overestimate
B16, an underestimate
C48, an overestimate
D28, an underestimate
E28, an overestimate

Q10:

The table gives sampled values of an increasing function . Use the data to give a lower and upper bound for .

	10	13	16	19	22	25
			1	4	8	10

Alower bound: , upper bound: 1
Blower bound: , upper bound: 20
Clower bound: , upper bound: 3
Dlower bound: , upper bound: 60
Elower bound: , upper bound: 66

Q11:

Calculate the left endpoint estimate of with subintervals. Is the result an overestimate or underestimate of the actual value?

A48, an underestimate
B18, an underestimate
C16, an underestimate
D16, an overestimate
E48, an overestimate

Q12:

Calculate the right endpoint estimate of with subintervals. Is the result an overestimate or underestimate of the actual value?

A16, an overestimate
B13, an underestimate
C48, an underestimate
D16, an underestimate
E48, an overestimate

Q13:

Suppose that and . Find two estimates of to three decimal places using the left and then the right endpoint methods with 8 rectangles.

A7.843, 7.686
B0.7689, 0.686
C7.769, 7.843
D7.769, 7.686
E7.686, 7.769

Q14:

Calculate the right endpoint estimate of with 8 subintervals of equal width. Give your answer to 2 decimal places. State whether your estimate is an overestimate or underestimate of the actual value of the integral.

A2.77, an underestimate
B2.77, an overestimate
C4.78, an overestimate
D3.61, an underestimate
E4.78, an underestimate

Q15:

Calculate the left endpoint estimate of with 8 subintervals of equal width.