2nd PUC Basic Maths Question Bank Chapter 21 Definite Integral and its Applications to Areas

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Karnataka 2nd PUC Basic Maths Question Bank Chapter 21 Definite Integral and its Applications to Areas

2nd PUC Basic Maths Definite Integral and its Applications to Areas One Mark Questions and Answers

Question 1.
2nd PUC Basic Maths Question Bank Chapter 21 Definite Integral and its Applications to Areas 1
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Question 2.
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Question 3.
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2nd PUC Basic Maths Question Bank Chapter 21 Definite Integral and its Applications to Areas

Question 4.
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Question 5.
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Question 6.
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Question 7.
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Question 8.
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2nd PUC Basic Maths Question Bank Chapter 21 Definite Integral and its Applications to Areas

Question 9.
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Question 10.
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Question 11.
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2nd PUC Basic Maths Question Bank Chapter 21 Definite Integral and its Applications to Areas

Question 12.
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Question 13.
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Question 14.

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Question 15.
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2nd PUC Basic Maths Question Bank Chapter 21 Definite Integral and its Applications to Areas

Question 16.
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Question 17.
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Question 18.
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Question 19.
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2nd PUC Basic Maths Question Bank Chapter 21 Definite Integral and its Applications to Areas

Question 20.
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Question 21.
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Question 22.
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2nd PUC Basic Maths Question Bank Chapter 21 Definite Integral and its Applications to Areas 44
= 3 + log(3 + 2) – 0 +1og(0 + 2) = 3 + log 5 – log 2 = 3 + log \(\frac { 5 }{ 2 }\)

2nd PUC Basic Maths Question Bank Chapter 21 Definite Integral and its Applications to Areas

Question 23.
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Question 24.
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Question 25.
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2nd PUC Basic Maths Question Bank Chapter 21 Definite Integral and its Applications to Areas

Question 26.
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2nd PUC Basic Maths Question Bank Chapter 21 Definite Integral and its Applications to Areas 53

Question 27.
If marginal cost = x2 – x – 2. Find the total cost obtained from an output of 40 units
Answer:
2nd PUC Basic Maths Question Bank Chapter 21 Definite Integral and its Applications to Areas 54

Question 28.
If the marginal revenue is given by f1(R) = 15 – \(\frac{x}{15}\) find the Total revenue obtoined from an output of 30 units
Answer:
2nd PUC Basic Maths Question Bank Chapter 21 Definite Integral and its Applications to Areas 55

2nd PUC Basic Maths Question Bank Chapter 21 Definite Integral and its Applications to Areas

Question 29.
If the marginal cost of a firm is f1(x) 10 + 6x – 6x2 where xis the output find the total cost among that the fixed cost is ₹ 125.
Answer:
TC = ∫MC dx
∫10 + 6x – 6x2 dx
= 10x + \(\frac{6 x^{2}}{2}-\frac{6 x^{3}}{3}\) + C given C = 125 = fixed cost
= 10x + 3x2 – 2x3 + 125

Question 30.
If the M R is f1(x) = 20 Find the total revenue & average revenue obtained from an output of 30 units
Answer:
2nd PUC Basic Maths Question Bank Chapter 21 Definite Integral and its Applications to Areas 56
2nd PUC Basic Maths Question Bank Chapter 21 Definite Integral and its Applications to Areas 57

Question 31.
Compute the total cost for the marginal cost function f1(x) = 3x2 – 6x + 12 assuming that the fixed cost is ₹500 Also find Average cost
Answer:
T C ∫MC dx
Where fixed cost C 500 Rs.
= ∫(326x + 12)dx = 3\(\frac{x^{3}}{3}\) – 6x\(\frac{x^{2}}{2}\) + 12x + C = x3 – 3x2 + 12x + 500
Ave cost = \(\frac{\mathrm{TC}}{x}\)
\(\frac{x^{3}-3 x^{2}+12 x+500}{x}\)
x2 – 3x + 12 + \(\frac{500}{x}\)

Question 32.
If the M R = 16 – x2 find the maximun total Revenue Also find the total & the average reveues also write the demand function
Answer:
TR ∫MR dx
= ∫(16 – x2)dz
l6x – \(\frac{x^{3}}{3}\) + C when output = 0,
TR = 16x – \(\frac{x^{3}}{3}\) Total revenue = 0 ∴ C = 0
Average revenue = \(\frac{\mathrm{TR}}{x}=\frac{16 x-\frac{x^{3}}{3}}{x}=16-\frac{x^{2}}{3}\)
Demand function = 16 – \(\frac{x^{3}}{3}\)

Question 33.
The marginal cost function is f1(x) = 3x2 + 2x + 1 wherex is the level of output find the total cost, ave cost, total variable cost, average variable cost given that fixed cost 70
Answer:
TC ∫MCdx
= ∫(3x2 + 2x + 1)
= 3 \(\frac{x^{3}}{3}\) + 2\(\frac{x^{2}}{2}\) + x + C given fixed cost C = 70
= x3 + x2 + x + 70
Total variable cost = 2x3 + x2 + x
Average cost = \(\frac{\text { Total cost }}{x}\)
= \(\frac{x^{3}+x^{3}+x+70}{x}\) = x3 + x + 1 + \(\frac{70}{x}\)
Average variable cost = 2x3 + x + 1

2nd PUC Basic Maths Question Bank Chapter 21 Definite Integral and its Applications to Areas

Question 34.
Find the area bounded by the curve y = x2, x – axis and the ordinates are x = 0, x = 1
Answer:
2nd PUC Basic Maths Question Bank Chapter 21 Definite Integral and its Applications to Areas 58

Question 35.
Find the area bounded by the curve y2 = 8x, x – axis & the lines x = 0, x = 2
Answer:
2nd PUC Basic Maths Question Bank Chapter 21 Definite Integral and its Applications to Areas 59

Question 36.
Find the area bounded by the curve x2 = 8y, y – axis and abscissas y = 3. y = 6 ∵ x = \(\sqrt{8 y}=\sqrt{8} \cdot \sqrt{y}\)
Answer:
2nd PUC Basic Maths Question Bank Chapter 21 Definite Integral and its Applications to Areas 60

2nd PUC Basic Maths Question Bank Chapter 21 Definite Integral and its Applications to Areas

2nd PUC Basic Maths Definite Integral and its Applications to Areas Five Marks Questions and Answers

Question 1.
Find the area between the curves y2 = 4x & x2 = 4y
Answer:
Given y2 = 4x but y = \(\frac{x^{2}}{4}\)
\(\left(\frac{x^{2}}{4}\right)^{2}\) = 4x
\(\frac{x^{4}}{16}\) = 4x ⇒ x4 – 64x = 0 ⇒ x(x3 – 43) = 0
⇒ x = 0 or x = 4
Required Area A = |A1 – A2|
2nd PUC Basic Maths Question Bank Chapter 21 Definite Integral and its Applications to Areas 61

Question 2.
Find the area bounded by the curve y2 = 5x and the line y = x
Answer:
y2 = 5x and y = x
⇒ x2 = 5x
⇒ x2 = 5x
⇒ x2 – 5x = 0
⇒ x(x – 5) = 0 ⇒ x = 0 or x = 5 are the limits
Required Area A = |A1 – A2|
2nd PUC Basic Maths Question Bank Chapter 21 Definite Integral and its Applications to Areas 62

2nd PUC Basic Maths Question Bank Chapter 21 Definite Integral and its Applications to Areas

Question 3.
Find the area between the curves x2 = 5y & y = 2x.
Answer:
Given x2 = 5y & y = 2x
∴ x2 = 5 (2x)
x2 = 10x ⇒ x(x – 10) = 0 ⇒ x = 0 or x = 10
RequiredArea = A = |A1 – A2|
2nd PUC Basic Maths Question Bank Chapter 21 Definite Integral and its Applications to Areas 63

Question 4.
Find the area bounded by the parabolas x2 = y & y2 = 8x
Answer:
Given y2 = 8 & y = x2
∴ (x2)2 = 8x
x4 – 8x = 0
⇒ x(x3 – 23) =0 ⇒ x = 0 or x = 2
RequiredArea A = |A1 – A2|
2nd PUC Basic Maths Question Bank Chapter 21 Definite Integral and its Applications to Areas 64

 

2nd PUC Basic Maths Question Bank Chapter 21 Definite Integral and its Applications to Areas

Question 5.
Find the ares exciosed between the parabola y2 = x and the Iine x + y = 2
Answer:
Given y = 2 – x & y2 = x
⇒ (2 – x)2 = x
4 + x2 – 4x – x = 0
x2 – 5x + 4 = 0
x = 4 or 1 (both are +v)
If x = 4, y = 2 – x = 2 – 4 = 2
If x = 1, y = 2 – x = 2 – 1 = 1
2nd PUC Basic Maths Question Bank Chapter 21 Definite Integral and its Applications to Areas 65

Question 6.
Find the area between the parabolas y2 = 4x & the line y = x
Answer:
Given y2 = 4x & y = x
⇒ x2 = 4x
⇒ x(x – 4) = 0 ⇒ x = 0 or x = 4
Required area A = A1 – A2

2nd PUC Basic Maths Question Bank Chapter 21 Definite Integral and its Applications to Areas 66
2nd PUC Basic Maths Question Bank Chapter 21 Definite Integral and its Applications to Areas 67

2nd PUC Basic Maths Question Bank Chapter 21 Definite Integral and its Applications to Areas

Question 7.
Find the area between the cures y = 8 – x2 & y = x2
Answer:
Given y = 8 – x2 & y = x2
x2 = 8 – x2
2x2 = 8 ⇒ x2 = 4 ⇒ x = ±2
2nd PUC Basic Maths Question Bank Chapter 21 Definite Integral and its Applications to Areas 68

2nd PUC Basic Maths Question Bank Chapter 21 Definite Integral and its Applications to Areas

Question 8.
Find the area bounded by the parabola y2 = 4ax & its latus rectum
Answer:
Required Area = 2 Area OAB
2nd PUC Basic Maths Question Bank Chapter 21 Definite Integral and its Applications to Areas 69
2nd PUC Basic Maths Question Bank Chapter 21 Definite Integral and its Applications to Areas 70
2nd PUC Basic Maths Question Bank Chapter 21 Definite Integral and its Applications to Areas 71

Question 9.
Find the arca exclosed between the parabola x2 = 4y and the line x = 4y – 2
Answer:
2nd PUC Basic Maths Question Bank Chapter 21 Definite Integral and its Applications to Areas 72
2nd PUC Basic Maths Question Bank Chapter 21 Definite Integral and its Applications to Areas 73

2nd PUC Basic Maths Question Bank Chapter 21 Definite Integral and its Applications to Areas

Question 10.
Find the area exciosed between y = 11x – 24 – x2 & the line y = x
Answer:
Given y = 11x – 24 – x2 & y = x
⇒ x = 11x – 24 – x2 ⇒ x2 – 10x + 24 = 0
⇒ (x – 4)(x – 6)
⇒ x = 4 or 6
2nd PUC Basic Maths Question Bank Chapter 21 Definite Integral and its Applications to Areas 74
2nd PUC Basic Maths Question Bank Chapter 21 Definite Integral and its Applications to Areas 75

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