← Calculus I

Unit 7: Antiderivatives and the Definite Integral

Running differentiation backward, then building the definite integral from scratch out of thinner and thinner rectangles.

10 topics · 35 questions

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4.10 Reversing Differentiation: Antiderivatives

  1. How it worksHow does asking what function differentiates to f(x) turn differentiation rules into antiderivative rules?
  2. Worked exampleFind the general antiderivative of f(x) = 4x3 - 6x + 5.
  3. Common mistakeWhy isn't there only one correct antiderivative for a given function?
  4. Worked exampleFind the general antiderivative of f(x) = 1/square root of x.
  5. Going furtherWhat happens to a family of antiderivatives when you graph several of them for different values of the constant C?
Your turn1 of 2
Find the antiderivative F(x) of f(x) = 6x3 - 6x + 8 that satisfies F(0) = 0.

4.10 Antiderivatives of Trigonometric and Exponential Functions

  1. How it worksHow do you reverse each transcendental derivative rule to build a matching antiderivative rule?
  2. Common mistakeWhy isn't the antiderivative of 1/x just ln(x) with no restriction on x at all?
  3. Worked exampleFind the general antiderivative of f(x) = sec2(x) - 4cos(x).
  4. Worked exampleFind the general antiderivative of f(x) = 3sin(x) + 2ex.
Your turn1 of 2
Find F(x), the antiderivative of f(x) = 4sec2(x) - 4cos(x), satisfying F(0) = 0.

4.10 Solving an Initial Value Problem

  1. How it worksHow does plugging a known point into a general antiderivative solve for the missing constant C?
  2. Worked exampleFind f(x) given that f'(x) = 6x2 - 4 and f(1) = 5.
  3. Worked exampleFind the position function s(t) given that s''(t) = -32, s'(0) = 48, and s(0) = 0.
Your turn1 of 2
Find f(x) given that f'(x) = 7x2 - 6 and f(1) = 14.

5.1 Approximating Area with Left, Right, and Midpoint Sums

  1. How it worksHow do left sums, right sums, and midpoint sums each pick a different height for the same rectangles?
  2. Worked exampleFind the right-sum approximation of the area under f(x) = x2 from 0 to 4 using 4 rectangles.
  3. Common mistakeWhy doesn't a left sum always underestimate the true area under a curve?
Your turn1 of 2
Find the right-sum approximation of the area under f(x) = 4x2 from 0 to 6, as a function of the number of rectangles n.

5.1 Watching the Rectangle Estimate Converge as n Grows

  1. Going furtherWhat happens to the gap between a left sum and a right sum as the number of rectangles doubles?
  2. How it worksWatch a Riemann sum's rectangles multiply from 4 to 8 to 40 to 200 while the total area locks onto one number.
  3. Worked exampleFind the midpoint-sum approximation of the area under f(x) = x2 from 0 to 4 using 8 rectangles, and compare it to the 4-rectangle estimate.
Your turn1 of 2
Find the midpoint-sum approximation of the area under f(x) = 4x2 from 0 to 6, as a function of the number of rectangles n.

5.1 The Trapezoidal Sum as an Average of Left and Right Sums

  1. How it worksHow does averaging a left sum and a right sum produce the exact same result as using trapezoids directly?
  2. Worked exampleFind the Trapezoidal Rule approximation of the area under f(x) = x2 from 0 to 4 using 4 subintervals.
  3. Common mistakeWhy doesn't the Trapezoidal Rule always give a better estimate than the Midpoint Rule?
Your turn1 of 2
Find the Trapezoidal Rule approximation of the area under f(x) = 4x2 from 0 to 6, as a function of the number of subintervals n.

5.1 Sigma Notation and Riemann Sums

  1. How it worksHow does sigma notation compress a Riemann sum's many rectangle areas into one compact expression?
  2. Going furtherWhat happens to a sigma sum's value as the number of terms n grows while the interval stays fixed?
  3. Worked exampleFind the Riemann sum for f(x) = x2 + 1 on the interval from 0 to 3 with n rectangles in sigma notation, and evaluate it for n = 6.
  4. Common mistakeWhy isn't the index variable inside a sigma sum the same kind of variable as x in the original function?
Your turn1 of 2
For the right Riemann sum of f(x) = 4x2 + 6 on [0,6] with n rectangles, find the area of the i-th rectangle, as a function of i.

5.2 The Definite Integral as a Limit of Riemann Sums

  1. How it worksHow does taking the limit of a Riemann sum as n goes to infinity turn it into the definite integral symbol?
  2. Common mistakeWhy doesn't a definite integral require the function being integrated to be continuous everywhere?
  3. Worked exampleFind the definite integral that the limit of a Riemann sum for f(x) = square root of x on the interval from 1 to 4 represents.
Your turn1 of 2
The limit as n approaches infinity of the Riemann sum (sum from i=1 to n of square root of (1 + 7·i/n))·(7/n) represents a definite integral of square root of x from 1 to what upper limit?

5.2 Properties of the Definite Integral

  1. How it worksHow do the properties of adding integrals over adjacent intervals and pulling out constants come straight from the sum they came from?
  2. Common mistakeWhy isn't the integral of a product of two functions the product of their two separate integrals?
  3. Worked exampleFind the integral of 3f(x) - 2 from 0 to 5 using the properties of definite integrals, given that the integral of f from 0 to 5 is 10.
Your turn1 of 2
Given that the integral of f(x) from 1 to 5 is 14, find the integral of 6·f(x) - 8 from 1 to 5.

5.2 Signed Area and the Definite Integral Below the x-axis

  1. How it worksHow does area counted below the x-axis subtract from area counted above it inside one definite integral?
  2. Going furtherWhat happens to a definite integral's value when a curve crosses the x-axis exactly in the middle of the interval?
  3. Worked exampleFind the definite integral of f(x) = x from -2 to 1 and compare it with the total area the curve encloses.
  4. Common mistakeWhy isn't a definite integral always equal to the area trapped between the curve and the x-axis?
Your turn1 of 2
Find the definite integral of f(x) = x from -3 to 4.