Integration Lesson AB Calculus

Multiple Choice

Which of the limits is equivalent to the following definite integral?

Multiple Choice

$$\int_2^5\left(x^3+3\right)dx$$  as limit of a sum is equivalent to

• $$\lim_{n\rightarrow\infty}\sum_{i=1}^n\left[\left(2+\frac{3i}{n}\right)^3+3\right]\frac{1}{n}$$  

• $$\lim_{n\rightarrow\infty}\sum_{i=1}^n\left[\left(2+\frac{3i}{n}\right)^3+3\right]\frac{3i}{n}$$  

• $$\lim_{n\rightarrow\infty}\sum_{i=1}^n\left[\left(\frac{3i}{n}\right)^3+3\right]\frac{3}{n}$$  

• $$\lim_{n\rightarrow\infty}\sum_{i=1}^n\left[\left(2+\frac{3i}{n}\right)^3+3\right]\frac{3}{n}$$  

Multiple Choice

$$\int_0^{\pi}\cos xdx\$$  as limit of a sum is equivalent to

• $$\lim_{n\rightarrow\infty}\sum_{i=1}^n\left[\cos\left(\frac{\pi i}{n}\right)\right]\frac{i}{n}$$  

• $$\lim_{n\rightarrow\infty}\sum_{i=1}^n\left[\cos\left(\frac{i}{n}\right)\right]\frac{i}{n}$$  

• $$\lim_{n\rightarrow\infty}\sum_{i=1}^n\left[\cos\left(\frac{\pi i}{n}\right)\right]\frac{\pi}{n}$$  

• $$\lim_{n\rightarrow\infty}\sum_{i=1}^n\left[\cos\left(\frac{i}{n}\right)\right]\frac{\pi}{n}$$  

Multiple Choice

$$\lim_{n\rightarrow\infty}\sum_{i=1}^n\left[\left(\frac{5i}{n}\right)^2+\frac{5i}{n}+1\right]\frac{5}{n}$$  in integral notation would be 

• $$\int_0^5\left(x^2+x+1\right)dx$$  

• $$\int_5^6\left(x^2+x+1\right)dx$$  

• $$\int_0^1\left(\left(5x\right)^2+5x+1\right)dx$$  

• $$\int_0^{10}\left(\frac{x^2}{2}+\frac{x}{2}+1\right)dx$$  

Multiple Select

$$\lim_{n\rightarrow\infty}\sum_{i=1}^n\left[2+\sqrt{3+\frac{4i}{n}}\right]\frac{4}{n}$$  in integral notation would be 

• $$\int_3^7\left(2+\sqrt{x}\right)dx$$  

• $$\int_0^4\left(2+\sqrt{x}\right)dx$$  

• $$\int_3^7\left(2x+\sqrt{x}\right)dx$$  

• $$\int_3^7\left(2+\sqrt{3+x}\right)dx$$  

Multiple Choice

For a function that is strictly decreasing, a right hand Riemann Sum is which of the following:

• Overestimate
• Underestimate
• Exact Solution
• Unable to Determine

Multiple Choice

For a function that is strictly increasing, a right hand Riemann Sum is which of the following:

• Overestimate
• Underestimate
• Unable to Determine
• Exact Solution

Multiple Choice

Another word for 'integral' is .. .

• Constant
• Derivative
• Antiderivative
• Theorem

Multiple Choice

Based on the table, use a left Riemann sum and 4 sub-intervals to estimate the Area under the curve. (Choose the correct set-up.) 

• 5(3) + 1(4) + 2(5) + 1(7)
• 5(4) + 1(5) + 2(7) + 1(6)
• 5(3) + 6(4) + 8(5) + 9(7)
• 0(3) + 5(4) + 6(5) + 8(7)

Multiple Choice

The graph of f consists of line segments and a semicircle, and the graph of g consists of line segments as shown in the given graphs. Evaluate the integral using properties and geometry.

$$\int_2^6\left[f\left(x\right)-g\left(x\right)\right]dx$$

• -1

• 7

• 2

• 9

Multiple Choice

Which of the following is equivalent to:

$$\int_7^3h\left(\alpha\right)d\alpha+\int_3^9h\left(\alpha\right)d\alpha$$  

• $$-\int_3^7h\left(\alpha\right)d\alpha$$  

• $$-\int_7^9h\left(\alpha\right)d\alpha$$  

• $$\int_7^9h\left(\alpha\right)d\alpha$$  

• None of the following are equivalent to the given

Antiderivatives

​This is called an indefinite integral.

Multiple Choice

$$\int_{ }^{ }7\ dx$$  

• 0

• 7 + c

• 7x + c

• -7x + c

Multiple Choice

$$\int_{ }^{ }3x^2+4x\ dx$$  

• $$x^3+4x^2+c$$  

• $$x^3+2x^2+c$$  

• $$\frac{3}{2}x^3+4x^2+c$$  

• $$3x^3+4x^2+c$$  

Multiple Choice

∫ 4 dx

• 0

• 4t + C

• 4x + C

• 2x² + C

Multiple Choice

∫(4 - 18x)dx

• -18

• 4x - 9x²

• 4x - 9x² + C

• (4 - 18x)² /2 + C

Specific Antiderivatives

What function has the derivative?

Until you memorize these this is going to be the question

What function has the derivative?

Until you memorize these this is going to be the question

What function has the derivative?

Until you memorize these this is going to be the question

What function has the derivative?

Until you memorize these this is going to be the question

What function has the derivative?

Until you memorize these this is going to be the question

Multiple Choice

$$\int_{ }^{ }\left(\frac{1}{x}\right)dx$$  

• $$1+c$$  

• $$\ln\ \left|x\right|\ +c$$  

• $$-\frac{1}{x^2}+c$$  

• $$x^0+c$$  

Multiple Choice

$$\int_{ }^{ }\cos\left(x\right)dx$$

• sin(x)+c

• -sin(x)+c

• -cos(x)+c

• $$\cos^2\left(x\right)+c$$

Multiple Choice

$$\int_{ }^{ }\sin\left(x\right)dx$$

• cos(x)+c

• -cos(x)+c

• -sin(x)cos(x)+c

• $$\sin^2\left(x\right)+c$$

Multiple Choice

$$\int_{ }^{ }\sec\left(x\right)\tan\left(x\right)dx$$

• tan(x)+c

• sec(x)+c

• csc(x)+c

• -csc(x)+c

Multiple Choice

$$\int_{ }\sec^2\left(x\right)dx$$

• sec(x)+c

• tan(x)+c

• 2sec(x)+c

• -csc(x)+c

Multiple Choice

$$\int_{ }^{ }e^xdx$$

• $$ex$$

• $$e^x$$

• $$e^x+c$$

• $$\ln\left(x\right)+c$$

Multiple Choice

$$\int_{ }^{ }\frac{1}{2\sqrt[]{x}}dx$$

• $$\sqrt[]{x+c}$$

• $$\sqrt[]{x}+c$$

• $$\left(\sqrt[]{x}\right)^2+c$$

• $$\left(\sqrt[]{x}\right)^{\frac{1}{2}}+c$$

Match

Match the following

• $$\int_{ }^{ }\frac{1}{\sqrt[]{1-x^2}}dx$$

• $$\int_{ }^{ }\frac{-1}{\sqrt[]{1-x^2}}dx$$

• $$\int_{ }^{ }\frac{1}{1+x^2}dx$$

sin^-1 (x)+c

cos^-1 (x)+c

tan^-1 (x)+c

• sin^-1 (x)+c

• cos^-1 (x)+c

• tan^-1 (x)+c

Multiple Choice

What does "+C" represent in the equation for the antiderivative of a function f'(x)?

• An unknown constant

• The axis of symmetry

• The coefficient

• A trig value

Multiple Choice

∫ t⁶ dt

• 6t⁵
• 6t⁵ + C
• 1/7 t⁷ + C
• t⁷ + C

Multiple Choice

Identify the Integral notation symbol $$\int f\left(x\right)dx$$  

• Indefinite Integral

• Definite Integral

• Integration

• Differentiation

Multiple Choice

The process of finding $$\frac{\text{d}y}{\text{d}x}=f'\left(x\right)$$  is known as_____

• Integration

• Differentiation

• Limitation

• continuation

​The expression F(x)+C is called the general antiderivative of f since it generalizes the family of antiderivatives of f.