Differential and Integral Calculus Multiple Choice Questions on “Change of Order of Integration: Double Integral”.
1. Which of the following is not a property of double integration?
a) ∬ af(x,y)ds = a∬ f(x,y)ds, where a is a constant
b) ∬ (f(x,y)+g(x,y))ds = ∬f(x,y)ds+ ∬g(x,y)ds
c) (∬_0^Df(x,y)ds = ∬_0^{D1}f(x,y)ds+ ∬_{D1}^{D2}f(x,y)ds,) where D is union of disjoint domains D1 and D2
d) ∬(f(x,y)*g(x,y))ds = ∬f(x,y)ds*∬g(x,y)ds
View Answer
Answer: d
Explanation: The following are the properties of double integration:
- ∬af(x,y)ds = a∬f(x,y)ds
- ∬f(x,y)+g(x,y))ds = ∬f(x,y)ds+ ∬g(x,y)ds
- (∬_0^Df(x,y)ds = ∬_0^{D1}f(x,y)ds+ ∬_{D1}^{D2}f(x,y)ds)
2. The region bounded by circle is an example of regular domain.
a) False
b) True
View Answer
Answer: b
Explanation: A domain D in the XY plane bounded by a curve c is said to be regular in the Y direction, if straight lines passing through an interior point and parallel to Y axis meets c in two points A and B. Hence, region bounded by circle is an example of regular domain.
3. What is the result of the integration (∫_3^4∫_1^2(x^2+y)dxdy)?
a) (frac{83}{6} )
b) (frac{83}{3} )
c) (frac{82}{6} )
d) (frac{81}{6} )
View Answer
Answer: a
Explanation: Given: (∫_3^4∫_1^2(x^2+y)dxdy)
Integrating with respect to y first, we get,
(∫_3^4(x^2(y)_1^2+(frac{y^2}{2})_1^2)dx= ∫_3^4(x^2+frac{3}{2}) dx)
Next integrating with respect to x, we get,
((frac{x^3}{3})_3^4+frac{3}{2}(x)_3^4= frac{37}{3}+frac{3}{2}=frac{83}{6})
4. Volume of an object expressed in spherical coordinates is given by (V = ∫_0^2π∫_0^frac{π}{3}∫_0^1 r cos∅ ,dr ,d∅ ,dθ.) The value of the integral is _______
a) (frac{√3}{2})
b) (frac{1}{√2} π)
c) (frac{√3}{2}π)
d) (frac{√3}{4} π)
View Answer
Answer: d
Explanation: Given: (V = ∫_0^2π∫_0^frac{π}{3}∫_0^1 r cos∅ ,dr ,d∅ ,dθ.)
( V = ∫_0^2π∫_0^{frac{π}{3}}(frac{r^2}{2})_0^1 cos∅, d∅, dθ)
( V = frac{1}{2} ∫_0^{2π}(sin∅)_0^frac{π}{3} d∅ ,dθ)
(V = frac{1}{2}×frac{√3}{2} ∫_0^2π dθ)
( V = frac{1}{2}×frac{√3}{2} ×2π)
( V = frac{√3}{2} π )
5. Which of the following equation represents Moment of Inertia of a plane region relative to x-axis?
a) ∬x2 f(x,y)dxdy
b) ∬xf(x,y)dxdy
c) ∬y2 f(x,y)dxdy
d) ∬yf(x,y)dxdy
View Answer
Answer: c
Explanation: Moment of Inertia of a plane region,
Relative to x-axis is given by,
Ixx=∬y2 f(x,y)dxdy
Relative to y-axis is given by,
Iyy=∬x2 f(x,y)dxdy
6. What is the mass of the region R as shown in the figure?
” alt=”” width=”360″ height=”311″ data-src=”2020/06/integral-calculus-questions-answers-change-order-integration-double-integral-q6″ data-srcset=”2020/06/integral-calculus-questions-answers-change-order-integration-double-integral-q6 360w, 2020/06/integral-calculus-questions-answers-change-order-integration-double-integral-q6-300×259 300w” data-sizes=”(max-width: 360px) 100vw, 360px” />
a) 8
b) 9
c) (frac{9}{2} )
d) (frac{9}{4} )
View Answer
Answer: b
Explanation: From the above figure, we can see that X-axis ranges from 0 to 3 and Y-axis ranges from 0 to 2.
Therefore, the mass of the region is given by,
(M = ∫_0^2∫_0^3xy ,dx,dy)
( = ∫_0^2y(frac{x^2}{2})_0^3 dy = frac{9}{2}(frac{y^2}{2})_0^2 = 9 )
7. The half-interval method in numerical analysis is also known as __________
a) Newton-Raphson method
b) Regula Falsi method
c) Taylor’s method
d) Bisection method
View Answer
Answer: d
Explanation: The Bisection method, also known as binary chopping or half-interval method, is a starting method which is used, where applicable, for few iterations, to obtain a good initial value.
8. Given (∫_0^8x^frac{1}{3}dx,) find the error in approximating the integral using Simpson’s 1/3 Rule with n=4.
a) 1.8
b) 2.9
c) 0.3
d) 0.35
View Answer
Answer: d
Explanation: Given: (∫_0^8x^frac{1}{3}dx,n = 3,)
Let (f(x)= x^frac{1}{3},)
(∆x = frac{b-a}{2}= frac{8-0}{2}=4) ………………since b=8, a=0 (limits of the given integral)
Hence endpoints xi have coordinates {0, 2, 4, 6, 8}.
Calculating the function values at xi, we get,
(f(0)= 0^frac{1}{3}=0)
(f(2)= 2^frac{1}{3})
(f(4)= 4^frac{1}{3})
(f(6)= 6^frac{1}{3})
(f(8)= 8^frac{1}{3} =2)
Substituting these values in the formula,
(∫_0^8x^frac{1}{3} dx ≈ frac{∆x}{3} [f(0)+4f(2)+2f(4)+4f(6)+f(8)])
( ≈frac{2}{3}[0+4(2^frac{1}{3})+2(4^frac{1}{3})+ 4(6^frac{1}{3})+2] ≈ 11.65)
Actual integral value,
(∫_0^8x^frac{1}{3} dx= left(frac{x^frac{4}{3}}{frac{4}{3}}right)_0^8=12)
Error in approximating the integral = 12 – 11.65 = 0.35
9. A sphere with the dimensions is shown in the figure. What is the error that can be incorporated in the radius such that the volume will not change more than 4%?
” alt=”” width=”197″ height=”195″ data-src=”2020/05/integral-calculus-questions-answers-change-order-integration-double-integral-q9″ />
a) 0.127%
b) 0.0127%
c) 12.7%
d)1.27%
View Answer
Answer: b
Explanation: We know that volume of the sphere is,
(V = frac{4}{3} πR^3 )
Differentiating the above equation with respect to R we get,
(frac{dV}{dR}= frac{4}{3} π×3R^2=4πR^2)
Since the volume of the sphere should not exceed more than 4%,
(dR=frac{dV}{4πR^2}=frac{0.04}{4π(5)^2}=0.000127)
Error in radius = 0.0127%
10. The x-coordinate of the center of gravity of a plane region is given by, (x_c=frac{1}{M}∬xf(x,y)dxdy.)
a) True
b) False
View Answer
Answer: a
Explanation: The coordinates (xc,yc) of the centroid of a plane region with mass M is given by,
(x_c=frac{1}{M} ∬xf(x,y)dxdy)
(y_c=frac{1}{M} ∬yf(x,y)dxdy)
Global Education & Learning Series – Differential and Integral Calculus.
To practice all areas of Differential and Integral Calculus, here is complete set of 1000+ Multiple Choice Questions and Answers.