Question

Draw a quick but accurate sketch of f(x) = √x²−4 over the interval [−4,0]. This covers the interval of integration.

1. Partition the interval of integration into 10 intervals. Show this on your graph with a right or left Riemann Sum

2. Create a table showing your interval index, i, the value x_i at which you evaluate f(x) in each interval, the values of f(x_i) and ∆x for each interval, and the contribution each rectangle makes toward the Riemann Sum. Evaluate the Riemann Sum for f(x) over the integration interval using your partition.

Answer 1

All the required formulae to solve this problem is:

The definite integral in terms of Raimann Sum is:

where,

, n is number of intervals

Also remember that,

where k is a constant

Read all the above formulas ones again and lets start solving the problem

The given function:

in interval [-4,0]

the plot of the above function is:

Tips to draw the plot accurately:

• According to the line spacing on the graph sheet select a suitable x values at regular intervels. In the above graph line spacing is 2.5mm, so I am selecting 5mm as interval.
• Now evaluate f(x) value for each x .
• plot the points (x,f(x)) on the graph.
• Join all the points with free hand to obtain thee cruve accurately.

Given that, the range of x is -4 to 0 with 10 intervals

so, the stepsize is:

The below table is prepared using above formulas

i			(contribution of each rectangle towards Raimann sum)
1	-3.6	2.9933	1.19732
2	-3.2	2.4979	0.99916
3	-2.8	1.9595	0.7838
4	-2.4	1.3266	0.53064
5	-2.0	0	0
6	-1.6	undefined	undefined
7	-1.2	undefined	undefined
8	-0.8	undefined	undefined
9	-0.4	undefined	undefined
10	0	undefined	undefined

The given problem is:

have to be solved using Raimann sum method

In the question it is given that n = 10,

= 1.19732 + 0.99916 + 0.7838 + 0.53064 + 0 = 3.51092

Hence, the Raimann sum of f(x) over the given interval is 3.51092.

Happy Learning!! Cheers!!

PhionexZura