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,iVBORw0KGgoAAAANSUhEUgAAAGwAAAAiCAYAAAC6NDzRAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsMAAA7DAcdvqGQAAAIMSURBVGhD7ZSLbcMwDAW9S4bJLBklk2SQzJFZXFzgB7yyUlzLdgGiPIBIbH1M8SRNc5GKEpaMEpaMEpaMEpaMEpaMEpaMEpaMEpaMEpaMEpaMEpaMQ4VN0zQ/Ho/l6SeXy2W+3W7L0/zuy5jX67W8GYPxzKNY436/v3PJyGHCVPzr9bq8+Q4Fol3C1J/YK8zn4D9CetBGn38vTKfnk4AzThhzrBWfPoT49yfs+Xy+T5aupt4OP0PYb4rPN0qYgQSkAeJ6xRgRxrz0UXjhmc/b4nWsDeQBEubtnhfwrDaf18eQm29OrUdxBrtnZQEuSElLoLNVmGQJ9XdpoyeMdxqneQU5en/PWzcJ8F7C6O9r4/9aXiPsFkbCvsuAxXvywhcOKlRPGMWJc8dC0L5WmJYwH6ONoTz4H0P9ySmeZOBda1xvbaPsFsZCWokSEfpuEUb/KIxnn/toYUQrd6G+CuXOfK1b5Wh2CYvXgNCivEgwIizu5ljso4UB/9eKL7EEtG6DM9glzBcZ0RXhbBWmdi8Ec3wqfgvmYAzBt9aEkWPMXXmzLvXjV/2Uq4vmO721jTIkTIkqvIAQ73PaKZCeWbwWqOgtLPbzb/mcRG+HKx++KxmKOL8KHteg95Id3wNtPkb5aEP08tvCrhNW/D0lLBklLBklLBklLBklLBklLBklLBklLBklLBklLBXz/AWaQIWifIfHMQAAAABJRU5ErkJggg==.
• C) data:image/png;base64,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.
• D) data:image/png;base64,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.

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

data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAeEAAAAjCAYAAACjHbIFAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsMAAA7DAcdvqGQAAAqGSURBVHhe7ZuLkdU4EEVJgRhIgRwIgRhIgQzIgAwoAiACEiABMiCH2TrsnK27vZLs9/F42OlTpRpbllr9k9p+7L56aJqmaZrmFLoIN03TNM1JdBFumqZpmpPoItw0TdM0J9FFuGmapmlOootw0zRN05xEF+GmaZqmOYkuwk3TNE1zEl2Em6ZpmuYkugg3TdM0zUk8qyL8/fv3h/fv3z/8+PHjsWfNp0+fHj58+PDw8+fPx56m+S/k1atXr/5p3B/Ju3fv/rVe0zTNjOUJQZHLw+TNmzePT/4mD7f6rPLr16/fBdPx3759e3zyN1++fHl4+/bt73GXgA6svbdwPzWXvlgwFr9f4odL4nAPXI+/QuGhPUdG+t6Ke+P169ePPf/FMUdBrFnjKJDPnr0V/f+nvyzj66fYX88N9vU98uBPwr1rO3Kf7TohUIIimbCh6M/rOiYhkBQYr/NA5JrD7NICLMxnc1w7/yiufbH4/Pnz7nmXxuFWPFBpxpB4cn9EEf748eNvX9yCOqvvvUA3c3rEkUWYfEf2kYfDPch8+ZOLsLF8aUXYvf2SijA5m/ZynuKDo87Vq4swSbn30HUjzjYhcig8t8BB/ZwOJGy+5cUCW1YHvFwSh3thPPkr6HCEHsjc44cVI33vwVbeenAfBQXh2pxnLx55sGYu6P8/uQgDvr6lCB+xP44AO+vefmlFuO7bW/baFlcXYYKyN6k4RGfJ69fbrRuUw/A5vaVuHdBbULzxy1bhuCQO98IkrRv1qfXYy0jfe8BL1krmcy7CxOqog7XmpP5/yUXYr6nnjudx3dsvqQhX7lWjZlxVhLnPtgpQHVu/albJyc99POMv8O+qbILRem701ZenOqzankMNPbCD8R6Eeeisgoa9HN7+xIq+zB1tbta4xLc5dtYP9FkgaCM987njRV/XjZo+qBg7bHc94zta3zVoxrTK0Je0+t8YJCN9Abn4hvk8Rx9sWOWQoIu6MYfrmtv6cA/pb67TfnMj+8Dcy7lpI7bZT3MPMy/7q1+EZ8gW5fG35pToCxtrqjcxc21tEs8B24qqf+YP9+kPn6VeqXtdd2YXIJe1c076B/QRzf2Q+tByDA1SP+Nkn2uknL3+w37uiUGuMSLzi+YazEPnXEMdJWNSfVJJPbgGfeKauZYxRC7Pq/9s4r06znwD6s0Y5I6gf/bsHoyjUUiDBKV04BYYOvsqxLEjORxyzNGB3DNudkCaQLMD5V6QEBza2I8uNNb1RQHUucJYxrkxKBzYhG0j8M3Wv4eO4pB+cC2TiGuayT4iNy64+WXka3QYxVGwG/tZF3noU3OqUu2nyKW/zamtf4oY6QvMo7nJ0X/L38LarotOozWq37ZQRsJ9+sk4ggeItqO/z5WlbfRnzHPsCOambOQ5fpbfwrjMBXVRn5pfyEtdqq4J+uQzrp3rGnUuurgWMEa7uNa/2qXPKsaz+sH59HsNqZtjE/RSD6i2obPyeJZjmauPGeM6wDVy9LNNH3CdspIaG2Ad+lyDv64NrgXOTz8kzFNOHTuyn+eM4xnXI9nMSX0Zrz4z3wAylVVtSlKnI9h1QowMXymd6Oh0UoIjVnI8dD2AZxiw2Tr3Aj2qvqybX2JbNoEFYFaAwcRbUeMwWpvYIcfE5LrGcwbyGZ96jHzNmls2A8WTpN6zPmvnyw2YT/lStyVvpC/zeZnKnCK2db0ZjCV+Ob+usSd+FcanLdzX+Ap25/1sT9KHnDxM6GP8CuYof+TDGVUP55p/wL12ql9tOX6E49IO7tN/5ktt1U/GijZblzH1QEaOslK+zfHoxH1S5amDPtb3wLiUa4OV/7Q/44aslJ2MxiM/fZx6G9vaRrnl2BnVH443HvW50J/2ZPxXvmEO1yuQlb44gl0nBIqmYVA32oxR8iU4Yuvrwy+OFQZs5TCeb7WtdRiTvsifJQUZW77hsN8asydJahxGcam+qTaMQI7jagxHvmbNLXuAl5WUtYINN/pf2bJ4emi4UUfM9K0HxdbP2kkd6xr5UrUnfpWMX/oe+7hOG+phWmPPtXYzLg+wkf0V5qZ85dFWc6se+iZjxD32AHqlXVuoA1Q7Ui4gd3RwizHi70jPpPoQtNU8nGEcE+doO/LTnmrXTK+V/+oawPiMazIaX32cfsCujPWKkQ+S6t8aj5H/QZ0lbVv5Bphny7yRUd+9mXskGClo8m0x+nJMtgLDcxyJnBUGbPW1fCuuURO0vkSQBKsXCw5qnnOQr0DOyjdQ48B9nVOTmetVctWY1PuZH7bygdgQR+aufgEAN1aNJ/ZlLqDb1kvcSF/uc7PyZUzfHkYvXqP9sCd+lYwVMoH8R1bqC/ZL6kB/Hlj1nnHKn1F9JMybPYPqi7RJuDcHGTuTVal61/uUC+ZRri1Vr5GeSfUhpO7MzRxL6h4S9ec5cx2HDmkH6+Z9svKf9qdeNW+S0fjq4/SDPtuDY/f6t46vzxN9gI9S/so3CXKr3bBn7q3s8h7K1QSoG20GhWZliEEfBYY1WcevJw69TIZkFaB7oa4c2OjC1ywFoeqE3owbwQGO35SFbeg+sp/+rQJT46Dc7GNM6sjz2YYG9Vcn5qY9bo5MWNZb5QP+4rkvIPiO+TM91KHGnBjnT9G+5CE3+5ORvsixmDPP/DGuQF/aLRZsXyTqvYzmc0//CvTAZvVVt/QD0JeyeG4M6tr0M164ZzwxnsUgdUWXXJ/rmR1VD/2fOc696xrrjA/zRnsCucoG5qVeKVeq7cAc9Uo/cz9aF+pz9RZk5j2oW45Nv9Ff48g4+lKPkW5V9sh/tPqsrpfkeMeYK0J/+pPrjAnk+ISxORccq42CTO5p6FXXTfRjXXflG/7yXJif456Kf2dMQafYdED20dKQhEOJ51uG4WzWEhPBw5CDkbUp6PWgEw72lHEU6IRu6Iwu6FXtV3/+JoxlHvaAX4Uz/zF2lszA3GzKcX2bMmr/yl/omvO9/vr16z/XNGKbm4XrCvORV4vWyjbkEm/i6jz1zxxQFr7UrxVkMSbzEF8hH714EUI29xkf/ENfBb2Zr4+YM8pL90/CPTqv0F+i3ak/9zbGZwycW8d4DeqW6ySOpTGWtVPGKM6CbxyX16N7c1B9an/FWNrUyVjY6vz6HJ9C+i3tS18nqefIdykv5eTeS9n24xcxnpXUr84Z+a/6qvoeeSP0FTLSb+hUdZDsy/4RNRZJ9msTPpqtm9CfPpGRb+xPufaL80Yy78naWzfCYTM6xCokC+Nmh+gWHKIE9tr5e2ANXwrEpB6tS8LWoF4CMpE9Owya/XgYXepL4l1jfgluYiGm5LkFoGma5rAizIFDYdx7iFHQ+Oq5tJD6NTr6ErknFFV/vgRfHGb23fpigdxcr7mea4qwb9+3UIswMT06T5um+bM4pAh76HGIXVKEmEfh2XtQ8aXNGk/xZYFOvCRgF43Cv/Wle+2LxbXzmjHmo+2SYnwN9SfJpmmaGX1CHMylLxa8VFDcuwA3TdP8/+ki3DRN0zQn0UW4aZqmaU6ii3DTNE3TnEQX4aZpmqY5iS7CTdM0TXMSXYSbpmma5iS6CDdN0zTNSXQRbpqmaZqT6CLcNE3TNKfw8PAXnVOS5JrM/XsAAAAASUVORK5CYII=.

• A) The area is dependent on the width of the interval [a, b].
• B) The area is infinite.
• C) The area is undefined.
• D) The area is zero.

How does the axis of rotation impact the choice between using the disk method or the washer method?

• A) It has no affect on the choice
• B) It determines the shape of the solid
• C) It determines the thickness of the cross-sections.
• D) It affects whether the solid has holes or voids along the axis.

$\frac{{{x^4}}}{4} - \frac{1}{4} = \_\_\_\_\_.$

• A) $\int\limits_1^x {{t^3}dt}$
• B) $\int\limits_1^x {{t^4}dt}$
• C)
• D)

Which of the following statements is true about $\int\limits_0^1 {(\sin x\cos x)dx}$?

• A) $$\int\limits_0^1 {(\sin x\cos x)dx} = \frac{1}{2}\int\limits_0^1 {\sin 2xdx}$$
• B) $$\int\limits_0^1 {(\sin x\cos x)dx} = \int\limits_0^1 {\sin xdx} + \int\limits_0^1 {\cos xdx}$$
• C) $$\int\limits_0^1 {(\sin x\cos x)dx} = 2\int\limits_0^1 {\sin xdx} \times 2\int\limits_0^1 {\cos xdx}$$
• D) $$\int\limits_0^1 {(\sin x\cos x)dx} = \int\limits_0^1 {\sin xdx} - \int\limits_0^1 {\cos xdx}$$

Use cylindrical shells to find the volume of the solid generated when the region ‘R’ enclosed between $$y = 2x + 1$$ and $$y = - 2x - 3$$ in the interval [1,3] is revolved about the y-axis is ______.

• A) $$V = \int\limits_1^3 {2\pi x\left( {(2x + 1) - ( - 2x - 3)} \right)} dx$$
• B) $$V = \int\limits_1^3 {2\pi x\left( {(2x + 1) + ( - 2x - 3)} \right)} dx$$
• C)
• D)

$$Area\,\,of\,\,the\,\,\,region\,\,bounded\,\,by\,\,the\,\,curves\,\,y = x^2 + 2\,\,,\,\,y = \,\, - x\,\,\,\,;\,\,x = 0\,\,and\,\,x = 1\,\,is$$

• A) $$\frac{3} {{16}}$$
• B) $$None\,\,of\,\,these$$
• C) $$\frac{17} {{6}}$$
• D) $$\frac{5} {{16}}$$

In general, an antiderivative of a product is the product of the antiderivatives.

• A) True
• B) False
• C)
• D)