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Question To estimate the integral d x Using the left endpoint rule and right endpoint rule where n = 10 Getting the width H= b-a/n where n is the number of subinterval Ln=Rn= H= 0.4-0/10 =0.04Ln = (* 0.04 Calculating for the exponential function to get= (1+ 1.001 + 1.006 +1.0145 +1.0259 +1.041 + 1.059 +1.082 + 1.1078 + 1.138)0.04=0.419008Right endpoint ruleRn = (1.1 + 1.006 + 1.0145 + 1.0259 + 1.041 + 1.059 + 1.082 + 1.1078 + 1.1338 + 1.1735)*0.04=0.42578Question 2Using the midpoint rule Formula for solving integration using the midpoint rule, the midpoint of each subinterval is used.

The interval is first divided into two equal pieces.= Δx [f(x1) + f(x2)….. f (xn)]Where Δx = b-a/n and n is the number of interval Δx = 0.4 – 0 / 10Calculating for the values we get =0.04 ((Summing up and multiplying by 0.04= (1+ 1.001 + 1.006 +1.0145 +1.0259 +1.041 + 1.059 +1.082 + 1.1078 + 1.138 + 1.1735) * 0.04=0.4659Calculating for the exact value of the integral of the function is given by When integrated -1/2 [ ] +1/2 [ ]= 0.33676Question 3Calculating for the absolute errorAbsolute error = true value - Estimated valueAbsolute values for the left point rule = 0.

33676 – 0.419008 = -0.082248Absolute value for the right point rule = 0.33676 - 0.42578 = -0.08902Absolute value for the midpoint rule = 0.33676 – 0.4659 = -0.12914The absolute values of the three method are different this shows how the estimated values differ from the exact values. With only the left and right rule the approximation made is poor than the midpoint rule which is close to the exact value.

Question 4 The answer is guaranteed to improve if the numbers of subintervals are increased. As the number of subinterval increases the more the answer get closer to the exact value hence the answer will be more accurate when 100 subintervals are used instead of 10 subintervals.Question 5Calculating the equation below dx Integrating the above equation will not affect the exponential function but only the square root of x. Blackhouse and Cooper, constants, say C will be indicated at the end of the equation since it is not a definite integral.

= 2 + C Question 5bIntegral involving trigonometric functionHere the integral of sin x is first found and then multiplied by the integral 1/ cos3x.Integral of sine x = cos x To get the one for cos3x reduction formula is applied or integration by parts which gives = Sin x – 1/3 sin3x + cHence the final answer becomes=cos x + sin x – 1/3 sin3x + C Work CitedBackhouse and cooper. Pure mathematics. Hong kong: Longman house, 1988.