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Fundamental Theorem of Calculus:

The fundamental theorem of calculus has two statements. The first statement states that  if  f is a continuous function on the closed interval [a, b] and  A(x) be the area function. Then A′(x) = f(x), for all x ∈ [a, b].

And  the second statement states that  the function “f” is continuous on the closed interval [a, b], and  F is an indefinite integral of a function “f” on [a, b], then the second fundamental theorem of calculus is defined as: F(b) – F(a) = .

 

Fundamental Theorem of Calculus Sample Questions:

Question 1: Calculate
 

 

(a) 30.5

(b) 34

(c) 21

(d) 40.5


Answer: (a)

Explanation: Let I =

 

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Question 2: Find
 

10 sinx dx.

 

(a) -10

(b) 9

(c) 10

(d) -9


Answer: (c)

Explanation: I = 10 sinx dx

 

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Question 3: If  dx =  , find c such that f(c) equals the average value of f(x) =  x2   over  [0,2].

 

(a)

(b)

(c)

(d)


Answer: (c)

Explanation: We are looking for the value of c such that,

f(c)= dx  =   ()  =  

 

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Question 4: Find the derivative of


g(x)= 

 

(a)

(b)

(c)

(d)


Answer: (b)

Explanation: According to the Fundamental Theorem of Calculus, the derivative is given by g′(x)

 

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Question 5: Calculate


 dx=

 

(a)

(b)

(c)

(d)

Answer: (a)

Explanation: Let cos x=t
Differentiating w.r.t x, we get
sin x dx=dt

 

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Question 6: Find 


 20x3 dx.

 

(a) -63

(b) -75

(c) 73

(d) 75


Answer: (b)

Explanation: Applying the limits by using the fundamental theorem of calculus.

 

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Question 7: Calculate


2tanxdx.

 

(a) log 2

(b) log3

(c) 2 log2

(d) 0


Answer: (a)

Explanation: Apply the fundamental theorem of calculus.

 

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Question 8: Find
 

 

(a)

(b) 4e

(c) -4e

(d) None


Answer: (a)

Explanation: Apply the fundamental theorem of calculus.

 

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Question 9: Find


 = 

 

(a)

(b) 6 

(c)

(d)


Answer: (c)

Explanation: Apply the limits.

 

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Question 10: Find


=

 

(a)

(b)

(c) 

(d) 


Answer: (d)

Explanation: Apply the limits.

 

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