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Approximating Area and Distance:

In this method of approximation we approximate the area under the curve y = f(x) between the given interval. The area under the curve is approximated by constructing the rectangles under the curve of equal widths, the sum of areas of these rectangles gives the approximate area under the curve.

The area under the curve of a continuous function f(x) is given by:
 

The area under the curve of a continuous function

 

Approximating Area and Distance Sample Questions:

Question 1: Calculate the area of the curve using 4 rectangles f(x) =  x^2 +1 in the interval [0,2] using the left endpoint approximation.

 

(a) 3.05

(b) 4.67 

(c) 3.75

(d) 4.5


Answer: (c)

Explanation: Explanation 1 = Explanation 1.1 = 0.5

 

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Question 2: For an increasing function which end point method gives the over approximate result?

 

(a) Left endpoints

(b) Right endpoints

(c) Mid points

(d) Only endpoints


Answer: (b)

Explanation: If f(x) is increasing the right end points give the over approximate results.

 

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Question 3: Calculate the approximate area of the curve for which A^L = 7.56 calculated using left endpoints  and A^R = 9.12 calculated using right endpoints.

 

(a) 16.6666

(b) 8.34

(c) 7.98

(d) 17


Answer: (b)

Explanation: Use the approximate area formula..

 

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Question 4: Estimate the area of the function f(x) = x^3 using the four rectangles and right endpoints.

 

(a) 298

(b) 98

(c) 100

(d) 64


Answer: (c)

Explanation: A^r = 1f(1) + f(2) + f(3) +f(4)

 

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Question 5: For f(x) = x^3 in [0,4] calculate the area using four rectangles and midpoints.

 

(a) 36

(b) 34

(c) 62

(d) 64


Answer: (d)

Explanation: A^M = 1[f(0.5)+ f(1.5) + f(2.5)+ f(3.5)]

 

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Question 6:  Estimate the area of graph of f(x) = cos x from x= 0 to x = Question 6/2 using 4 rectangles.

 

(a) 0.7908 and 0.845

(b) 0.673 and 0.845 

(c) 0.845 and 0.7908

(d) 0.7908 and 1.1835


Answer: (d)

Explanation:  A^4 = Explanation 6(cos Explanation 6.1 + cos Explanation 6.2 + cos Explanation 6.3 + cos Explanation 6.4)

 

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Question 7: Find the expression for the area under the graph of f as a limit


f(x) = x^2 + Question 7 in [3, 10]

 

(a) Question 7 a

(b) Question 7 b

(c) Question 7 c

(d) none of above


Answer: (b)

Explanation: Explanation 7 = Explanation 7.2 = Explanation 7.3

 

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Question 8: We can improve the estimation of the area of the curve by

 

(a) Taking average of the areas

(b) By differentiation

(c) By increasing the number of rectangles

(d) By decreasing the number of rectangles


Answer: (c)

Explanation: By increasing the number of rectangles we can improve the estimation.

 

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Question 9: The area under the curve of a continuous function f(x) is

 

Question 9

Answer: (d)

Explanation: The area under the curve of a continuous function f(x)

 

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Question 10: If y= x^2, find the area of the curve for n rectangles using right endpoints.

 

(a) Question 10 a

(b) Question 10 b

(c) Question 10 c

(d) Question 10 d


Answer: (d)

Explanation: R^n = Explanation 10 + Explanation 10.1 + Explanation 10.2……. + Explanation 10.3

 

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