Integration - QUIZ 1 - Quiz & Trivia

1. $\int {(2{x}^{2}-5)}\, dx\, =\,$

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Explanation

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2. Integration is essentially the same as ..........

Differentiation

Antidifferentiation

Integration is essentially the same as antidifferentiation. Integration involves finding the antiderivative of a function, which is the reverse process of differentiation. Antidifferentiation allows us to find the original function when only its derivative is known. Therefore, integration and antidifferentiation are synonymous terms in calculus.

Explanation

Integration is essentially the same as antidifferentiation. Integration involves finding the antiderivative of a function, which is the reverse process of differentiation. Antidifferentiation allows us to find the original function when only its derivative is known. Therefore, integration and antidifferentiation are synonymous terms in calculus.

3. Find the equation of the curve, given that $\frac {dy} {dx}=3{x}^{2}+1$ and the point P (1,4) is on the curve.

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Explanation

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4. Find the area under the curve $y=3{x}^{2}+2x$ from x = 0 to x = 4

64 area units

80 area units

96 area units

108 area units

The area under the curve from x = 0 to x = 4 is equal to the integral of the function over that interval. Without knowing the specific function, it is not possible to calculate the exact area. Therefore, the given answer of "80 area units" is a possible area under the curve, but it cannot be confirmed as the correct answer without further information.

Explanation

The area under the curve from x = 0 to x = 4 is equal to the integral of the function over that interval. Without knowing the specific function, it is not possible to calculate the exact area. Therefore, the given answer of "80 area units" is a possible area under the curve, but it cannot be confirmed as the correct answer without further information.

5. $\int ^{3}_{0} {(10-{x}^{2})\, dx\, =}$

18

19

20

21

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Explanation

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6. $\int {{(2x-1)}^{10}}\, dx\, =$

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Explanation

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7. Given that $\frac {dy} {dx}={2x}^{4}-{3x}^{2}+x-2$ . Find the general expression of the function y.

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Explanation

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8. $\int {5\, dx}\, =$

Undefined

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Explanation

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9. Find the area of the shaded region.

20 area units

Area units

21 area units

Area units

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Explanation

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10. $\int ^{4}_{1} {(x-1)\, dx}\, =$

4

3

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Explanation

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11. Find the area enclosed between the curves $y={x}^{2}+7$ and $y={2x}^{2}+3$ .

Area units

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Explanation

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12. $\int {\frac {1} {\sqrt[{3}] {3x-2}}}\, \, dx\, =$

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Explanation

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13. $\int {\frac {{x}^{5}-2} {{x}^{3}}}\, dx\, =$

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Explanation

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14. The definite integral for a function that is below the x-axis is always positive. Is that true or false?

True

False

The definite integral for a function that is below the x-axis is not always positive. The value of the definite integral represents the signed area between the function and the x-axis. If the function is below the x-axis, the area will be negative. Therefore, the statement is false.

Explanation

The definite integral for a function that is below the x-axis is not always positive. The value of the definite integral represents the signed area between the function and the x-axis. If the function is below the x-axis, the area will be negative. Therefore, the statement is false.

15. This following improper integral has a finite value. Find its value. $\int ^{\infty }_{1} {\frac {2} {{x}^{2}}}\, dx$

0

1

2

Undefined

The given improper integral is not properly stated in the question, so it is difficult to provide a clear explanation. However, based on the answer provided, it can be inferred that the integral is convergent and evaluates to a finite value of 2. Without further information, it is not possible to determine the specific function or limits of integration involved in the integral.

Explanation

The given improper integral is not properly stated in the question, so it is difficult to provide a clear explanation. However, based on the answer provided, it can be inferred that the integral is convergent and evaluates to a finite value of 2. Without further information, it is not possible to determine the specific function or limits of integration involved in the integral.

16. $\int {\sqrt[{5}] {{x}^{2}}}\, dx\, =$

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Explanation

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17. Find the area of the region bounded by the graphs of $y=3x-1$ , y-axis, y = 1 and y = 5 lines.

Area units

6 area units

10 area units

To find the area of the region bounded by the graphs of y-axis, y = 1, and y = 5 lines, we need to find the area between the y = 1 and y = 5 lines. Since the region is bounded by the y-axis, the x-values will range from 0 to a certain value. To find this value, we need to find the x-intercept of the line y = 5. The x-intercept is where y = 0, so substituting y = 0 into the equation y = 5 gives us x = 0. Therefore, the x-values range from 0 to 0. The area of the region is then given by the formula A = (5-1) * (0-0) = 0. Thus, the area of the region bounded by the graphs is 0 area units.

Explanation

To find the area of the region bounded by the graphs of y-axis, y = 1, and y = 5 lines, we need to find the area between the y = 1 and y = 5 lines. Since the region is bounded by the y-axis, the x-values will range from 0 to a certain value. To find this value, we need to find the x-intercept of the line y = 5. The x-intercept is where y = 0, so substituting y = 0 into the equation y = 5 gives us x = 0. Therefore, the x-values range from 0 to 0. The area of the region is then given by the formula A = (5-1) * (0-0) = 0. Thus, the area of the region bounded by the graphs is 0 area units.

18. The area of the shaded region in the figure below can be found by calculating ......

A

B

C

D

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Explanation

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19. Find the volume of the solid figure generated by rotating the area bounded by $y=\sqrt {x}$ , y-axis, y = 0 and y = 2.

To find the volume of the solid figure generated by rotating the area bounded by the y-axis, y = 0, and y = 2, we can use the method of cylindrical shells. Since the area is bounded by the y-axis and two horizontal lines, we can consider a vertical strip of width dx at a distance x from the y-axis. The height of this strip will be 2, and the circumference will be 2πx. Therefore, the volume of this strip will be 2πx * 2 * dx = 4πx dx. Integrating this expression from x = 0 to x = 2, we can find the total volume of the solid figure.

Explanation

To find the volume of the solid figure generated by rotating the area bounded by the y-axis, y = 0, and y = 2, we can use the method of cylindrical shells. Since the area is bounded by the y-axis and two horizontal lines, we can consider a vertical strip of width dx at a distance x from the y-axis. The height of this strip will be 2, and the circumference will be 2πx. Therefore, the volume of this strip will be 2πx * 2 * dx = 4πx dx. Integrating this expression from x = 0 to x = 2, we can find the total volume of the solid figure.

20. Find the volume of the solid figure generated by rotating the area of the region bounded by $y={x}^{2}$ and $y=2x$ about the x-axis

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Explanation

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Integration - Quiz 1