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Questions and Answers
Given a table of values, which Riemann sum approximation uses the function's value at the midpoint of each subinterval to estimate the definite integral?
Given a table of values, which Riemann sum approximation uses the function's value at the midpoint of each subinterval to estimate the definite integral?
- Trapezoidal sum
- Midpoint sum (correct)
- Left-hand sum
- Right-hand sum
A bag initially contains 50 pounds of sand. Sand is added at a rate of $f(x)$ pounds per hour. Write an expression that represents the amount of sand in the bag at time $t = 20$ hours using an integral.
A bag initially contains 50 pounds of sand. Sand is added at a rate of $f(x)$ pounds per hour. Write an expression that represents the amount of sand in the bag at time $t = 20$ hours using an integral.
$50 + \int_{4}^{20} f(x) , dx$
When approximating a definite integral using a trapezoidal sum, the width of each subinterval must be uniform.
When approximating a definite integral using a trapezoidal sum, the width of each subinterval must be uniform.
False (B)
If $f'(x)$ represents the rate of snow being added to a slope and $M(t)$ represents the rate at which snow is melting, what does $\int_{0}^{6} (S(t) - M(t)) , dt$ represent?
If $f'(x)$ represents the rate of snow being added to a slope and $M(t)$ represents the rate at which snow is melting, what does $\int_{0}^{6} (S(t) - M(t)) , dt$ represent?
To evaluate $\int \frac{1}{4 + (x-3)^2} , dx$, you can use a trigonometric substitution based on the inverse tangent. The appropriate substitution involves identifying $a$ and $u$ such that $a^2 = 4$ and $u = $ ______.
To evaluate $\int \frac{1}{4 + (x-3)^2} , dx$, you can use a trigonometric substitution based on the inverse tangent. The appropriate substitution involves identifying $a$ and $u$ such that $a^2 = 4$ and $u = $ ______.
What is the correct antiderivative of $\int 2x^{-1} + 3x^{-2} , dx$?
What is the correct antiderivative of $\int 2x^{-1} + 3x^{-2} , dx$?
When finding the derivative of an integral with a variable upper limit using the Fundamental Theorem of Calculus, it is always necessary to apply the chain rule.
When finding the derivative of an integral with a variable upper limit using the Fundamental Theorem of Calculus, it is always necessary to apply the chain rule.
When using u-substitution to evaluate $\int x^3 \sqrt{x^4 + 3} , dx$, what should $u$ be?
When using u-substitution to evaluate $\int x^3 \sqrt{x^4 + 3} , dx$, what should $u$ be?
To rewrite the limit $\lim_{n \to \infty} \sum_{i=1}^{n} \ln \left(2 + \frac{5i}{n} \right) \cdot \frac{5}{n}$ as a definite integral, the corresponding integral would have limits of integration from 2 to ______.
To rewrite the limit $\lim_{n \to \infty} \sum_{i=1}^{n} \ln \left(2 + \frac{5i}{n} \right) \cdot \frac{5}{n}$ as a definite integral, the corresponding integral would have limits of integration from 2 to ______.
Match the function with its corresponding antiderivative:
Match the function with its corresponding antiderivative:
Flashcards
Midpoint Riemann Sum
Midpoint Riemann Sum
Approximate a definite integral using rectangles where the height is the function value at the midpoint of each subinterval.
Left Hand Riemann Sum
Left Hand Riemann Sum
Approximate a definite integral using rectangles where the height is the function value at the left endpoint of each subinterval.
Trapezoidal Sum
Trapezoidal Sum
Approximate the area under a curve by dividing it into trapezoids and summing their areas.
Snow melting Rate
Snow melting Rate
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Snow addition Rate
Snow addition Rate
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Fundamental Theorem of Calculus
Fundamental Theorem of Calculus
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Study Notes
- These are study notes covering several calculus topics
- The topics covered include Riemann sums, definite integrals, antiderivatives, the fundamental theorem of calculus, u-substitution, and setting up integrals
Riemann Sums
- Riemann sums approximate definite integrals using rectangles or trapezoids to estimate the area under a curve
- The width of each subinterval is denoted as h, and the height is determined by the function's value at a specific point within each subinterval
- Midpoint Sum: Uses the function's value at the midpoint of each subinterval as the height
- Left-Hand Sum Uses the function's value at the left endpoint of each subinterval as the height
- Right-Hand Sum: Uses the function's value at the right endpoint of each subinterval as the height
- Trapezoidal Sum: Averages the function's values at the left and right endpoints of each subinterval to determine the height
Definite Integrals
- Definite integrals calculate the net area under a curve between two limits of integration
- Properties and Theorems:
- The integral of a rate function f(x) gives the accumulation of the quantity f over the interval of integration
Antiderivatives
- Antiderivatives reverse the process of differentiation
- Inverse Trigonometric Functions:
- ∫ du/(a^2 + u^2) = (1/a) tan^-1(u/a) + C
- ∫ du/√(a^2 - u^2) = sin^-1(u/a) + C
Fundamental Theorem of Calculus
- Part 1: Connects differentiation and integration
- The derivative of the integral of a function from a constant to a variable x is simply the function evaluated at x
- When the upper limit of integration is a function of x, the chain rule must be applied
U-Substitution
- U-substitution simplifies integrals by changing variables
- Choose a substitution u that simplifies the integral, compute du, and rewrite all terms of the original integral in terms of u
Setting Up Integrals
- Integrals can express other math expressions
- Integrals provide a method for determining the area under a curve
- The area under a curve is determined by the integral and the limits of integration
- The following limit can be rewritten as an integral:
- lim (n→∞) Σ [ln(2 + (5i/n))] (5/n) = ∫ ln x dx
- a = 2
- b = 7
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