MCQ Bank
$\int\limits_a^b {f(x)dx} \, = \,\_\_\_\_\_\_\_\_.$
- A) $\int\limits_a^b {f(z)dz}$
- B) $\int\limits_b^a {f(x)dx}$
- C)
- D)
If the definite integral of f(x)=3 over [1,x] is greater than ‘12’ then -----
- A) x>5
- B) x>1
- C) x>3
- D) x>12
The volume of cylindrical shell for R = 2, r = 1 and h = 2 is -----------
- A) 5*pi
- B) 6*pi
- C) 3*pi
- D) 4*pi
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- A) data:image/png;base64,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.
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- C) data:image/png;base64,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.
- D) data:image/png;base64,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.
If the solid is revolved around the y-axis and generates a solid with a circular cross section of radius g(y) at y. Then the area of this cross section is
- A) $\pi \left[ {g(y)} \right]$
- B) $\pi {\left[ {g(y)} \right]^2}$
- C) $\pi r{\left[ {g(y)} \right]^3}$
- D) ${\left[ {g(y)} \right]^2}$
If f(x)= x and g(x)=x-1are integrable functions over the interval [a, b] for all xϵ[a, b], then which of the following expressions is true for f and g?
- A) $$\int\limits_a^b {f(x)dx} \leqslant \int\limits_a^b {g(x)dx}$$
- B) $$\int\limits_a^b {f(x)dx} \geqslant \int\limits_a^b {g(x)dx}$$
- C) $$\int\limits_a^b {f(x)dx} > \int\limits_a^b {g(x)dx}$$
- D) $$\int\limits_a^b {f(x)dx} < \int\limits_a^b {g(x)dx}$$
The integral of f(x)=sin(2x) from x=0 to x=pi is ........
- A) 2
- B) 1
- C) None of thes.
- D) 0
............ is used to prove the first fundamental theorem of calculus.
- A) Intermediate value theorem
- B) Mean value theorem for the derivatives
- C) None of these.
- D) Mean value theorem for the integrals
Which of the following alternative methods could be more efficient than slicing for finding the volume of certain solids?
- A) The disk method
- B) The shell method
- C) The cross-sectional method
- D) The washer method
Which of the following is the ‘mesh size’ in the partition: {[0,0.25],[0.25,0.75],[0.75,1.25],[1.25,2]} of the interval [0,2]?
- A) [1.25,2]
- B) [0.25,0.75]
- C) [0,0.25]
- D) [0.75,1.25]
If the value of definite integral of a function
f(x) taken from 1 to 3 is 2 and that of taken from 3 to 5 is 1 then value of definite integral taken from 1 to 5 is
- A) 0
- B) None of these
- C) 3
- D) 1
Constant of integration is taken to be ………. in definite integral.
- A) c
- B) k
- C) 0
- D) All of these
$$The\,\,bounded\,\,region\,\,between\,\,the\,\,parabola\,\,y = 4x^2 \,\,and\,\,the\,\,line\,\,y = 6x - 2$$
- A) $$\frac{1} {6}\,$$
- B) $$\frac{1} {3}$$
- C) $$\frac{1} {{12}}$$
- D) $$None\,\,of\,\,these$$
If a similar solid is rotated around the x-axis over a closed interval [a, b] then the corresponding volume of revolution is ....
- A) data:image/png;base64,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.
- B) data:image/png;base64,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.
- C) data:image/png;base64,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.
- D) data:image/png;base64,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.
Which statement accurately describes the comparison between the cylindrical shell method and the disk or washer method regarding their applications.
- A) Shell method is simpler to apply.
- B) Both methods can be used interchangeably.
- C) The disk or washer method is simpler to apply.
- D) The cylindrical shell method is always more accurate.
How can the volume of the solid formed by revolving a region R bounded by the graph of f(x) around the y-axis be approximated using cylindrical shells?
- A) data:image/png;base64,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.
- B) data:image/png;base64,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.
- C) data:image/png;base64,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.
- D) data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAQgAAABSCAYAAAColEraAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsMAAA7DAcdvqGQAAAk0SURBVHhe7ZwBkdw6EESPQjCEQjgEQjCEQhiEQRgEQRCEQAiEwXHY1Lt/XTWlr5FkS75d+/pVqXZty7I0mmlLsnefbsYYk2CBMMakWCCMMSkWCGNMigXCGJNigTDGpFggjDEpFghjTIoFwhiTYoEwxqRYIIwxKRYIY0yKBeKi/Pr16/bhw4fbp0+fbs/Pz7evX7/enp6eXraNGcUCcUEQhG/fvt1+//79IgqIw58/f15Eg21jRrG3XBgJAkIRt40Zxd5yYRhFxCkF258/f37dMqaPBeLCIAaIgkAsvn///rplTB8LxIVhOsG0Qmi68ePHj5d1CmN6WCAuihYo//79+7rnvxEETzZ+/vz5useYNhYIY0yKBcIYk2KBMMakWCCMMSkWCGNMigXihPCYkicURyRjIvaIk8JLUArqL1++vO7dTvwhlwXClNgjTgrvN/BOgwKbUcUMHz9+tECY/2GPODH68RUJsYgvRW2FV7AtEKbEHnFy+K2FRGLmvx705qUxEXvEyWENAWGQSPjHWGYlFogLwJ/BSCBI+v8HY2axQFyE+OiTBUf/WtOswAJxIXjcKZGYefRpjLBAXAhGDfHRp3/WbWaxQFwMPY0gzT76NGZKIOSIpPhfh9zJsmMl3OWYM5eOzMJbHDKrHC3A6e/T7jnXjk8PInqn4F6Lhbo+aebRZwZ9Rt9otEL/ne3pSelfrSlZ5qMr2OrH1IX6vtXocEogtDBWWxTTMQxQazz7OEbHlMfpPJyPV4B1TIbRX6ixH8NybfLfC4lE5N4CAaoXaWXwqm28f0EfkBRoK69zJPIv1Ze2xP/uFLQt89FVbPVjvWLfErSVTE8xqGxthEBAt4a4nJM1kv0YrIRrleXJuEd1YA912KOBnXSHJ60SK+xN0ES4VuYHjwj1rPlXSctHVzPix9HOb+VzhwkE+6TQJRoNZMbAsWtlZmBYRhv34C07ayuyM+loEc384NHQGk1txBDp+egR9PyYeOJ4OZo+kkMEgga0FLpnCA2PR6cOms7cYxRRCkQMynjXps3sIz/t0nnsp6Opu16bRiB7DjwKdlZ9jrob6s6W9Slt41gc0dRS9KNoo/hDNIKEfdh5KyovpnjNSMtHj6gb9PxYvtKyt2zN8djfnIf9szZlTAsEwRyNTAXVkBoYt2dEziUPDcrKiYyUeRRylIhEohzWYxcSwY+d6Gi21XFss5+O5XzaNYv6g/JII/bcigKD+tegvdiCusR2ydFr0x/sQX4+sQ9g11nhVBBmdYWePx1Vt9Z1FeCCuIvbgvZhT/kgZZLwKeq8lWmBIECiQOAscbtEHVRziggNxADkHTH8lnyjibb0qAkEbWNf2UbyEqwR6lzmlUD2bDSKHI8U73gr6AVTRHYRvTsmyBaUv2IERBllH5SM+ujqugHl1fwYMSIJ+Q32ryExVjzuEQdYIhBatEKVCeqWOutuMwLlaLrRc0DytITpKLYKRFnHmj2y8/eiflE/rQKno9zouC3Km8dIsMrRybfXySOU0wvmUR9dXTegvNJHdJ0oBhLm1k1Ro4wZP1oiEDImhse4LbYIBMg4pdFKRvIcQWy/eDSBkKO0hHsr9AvljooD0PboH6Pi0son+42IH+0n70ofXVU3Qf7SRxjR1ISUa7cEdkSAe4xHakIcIo8o6Rbji1pgldQMW4N8o6nnSMA1yRt5JIFQ/6xeeyAocPwtd85YDwVrb8qjqWYryLBp604qKGvErqM+urJuguuWPkL5tTLoA/LXphnsIx45vqWPSqYFIqrkiBO25ncoXjmVoHEtlRaUuaUjVkFncu1IFuDkLTv/SIFQOT3bbYU+pE/KEUnrbqX5us4ZaSNOLp8gb+bo0fcUFKWdQWLZY2QNYnXdBMejH2sqURMBXbsUWerCNXTuSFxmLBOIVqMjqnQpBIABMaiciE86oeaMkVaZR0N9uXZ0EHVcWR/aVwZR7S6QdfwWJKzUL3PePajcsm0SgAz8I7Y9BiE+VAaAnJxP9S/n8J38guMck38QXLS5LA/YT+qh62X+dETdoHZdjdRqqPzyOOdI3OgrYoi8e24UywSiFcAlOEqtsjSExlIeicaRr1e2nA0jvCXUTXWV86uTleQMtE37+A6qN4m2krc8P3PSHjiwylyJ+ruWuF4NiUfsc/oUm3EOdojgAxyLdZf9KCP2M2WTl/KoWyuoyzq0yHz0iLqJmh9jH/b1EtdAgMgfr6My8Yc9vjAtEHugAWrUCrLOfK8oiMvAuyKaNpDwgwxNaUYFd4WPjtZNPKIf30UgACXWnXQGggHVfOvRw6OiEcgK254B/Agf0Cglu0sqWLf4yayPjtYNHtWP7yYQGEIdsNcoGBXV3TN0uiLYEXuQHs3RjoA2Enhxvo0Q4A8EpWBbx7Yw46OjdYNH9uO7CYRgKEcnbB3KSZ3fQyCMgiP37lRXQndmofk2gSi/4PsecYjs8dGRusGj+/HdBcKsQQ6IsxmzCgvEIrjLzC4waSjMcHPL3UrnsVJtzEosEIvgzj0boEwRNEcdFRuGpgxTEYjZYSpleARiIhaIBRCYDPEJat35+STg+CSAy4Up4Lz42C2umI+KDddkaqHFsL1QT09RTIkFYgGIAwFGsPMdEAS+915QISAV3FEURgSC62nhaxatYdSEzLxfLBAL0HSAEUEcBRD4MXgJeoKwTBIDRhqiJxAaocRz9kI9KYu6zI5EzLWwQExCoGrUAApYxALhYDtOI0riCEJrEKWw1KDcUmhWJAuEiVggJtHQvAwyrT8Q9Nm8vlyD0Hm9pxiIR3nNVak1HTLvDwvECcmmKiuSMRF7hDEmxQJhjEmxQBhjUiwQJ0XrECvegQAWRXnq4helTMQCcWIQiNkXm3iSgsggODxBsUCYiAXipBDYCASf3P31FGJPgOvdB0TCAmEiFoiTwsiB9yUEL07xzgWCAQS9RKOWakJggTAlFoiTwrSANQNebOL7ijcgLRCmxAJxUuKr1votyCwWCFNigTghWn9g1MDogcXFEk8xzAosECeE9QeCHJhi8J19/K6j9RuOFpyH0MRfoxpjgTgheiwpCGqCO/6qdAtxZKHkkYQBC4QxJsUCYYxJsUAYY1IsEMaYFAuEMSbFAmGMSbFAGGNSLBDGmBQLhDEm4Xb7B9x8sdR5KUfGAAAAAElFTkSuQmCC.
Which of the following statements is true about $\int\limits_0^1 {\sec x\tan xdx}$?
- A) $$\int\limits_0^1 {\sec x\tan xdx} = [\sec x]_0^1 \times [\tan x]_0^1$$
- B) None
- C) $$\int\limits_0^1 {\sec x\tan xdx} = [\sec x]_0^1$$
- D) $$\int\limits_0^1 {\sec x\tan xdx} = [\sec x]_0^1 + [\tan x]_0^1$$
The value of $\int\limits_0^{\frac{\pi }{4}} {{{\tan }^2}x\,dx} \,\_\_\_\_\_\_.$
- A) None of the above
- B) $$1-\frac{\pi }{4}$$
- C) 0
- D) $$\frac{\pi }{4}-1$$
$$\begin{gathered} {\text{If the upper and lower limits for the definite integral are the same, then }} \ \int_a^a {f(x)} dx = \\ \end{gathered}$$
- A) negative integer
- B) none of these
- C) positive integer
- D) zero