AP Calc BC [Unit 8]

find the intersection of the two curves f(x) = 14x + x^2 - 2x^3 g(x) = x^2 - 4x

(-3, 21) (0, 0) (3, -3)

Average Value of a Function on an Interval

(1 / (b - a)) ∫ f(x) dx (from a to b)

Let f be the function given by f(x) = (x² + x) cos(5x). What is the average value of f on the closed interval [2, 6]?

-1.848

disc

formed by revolving a rectangle around an axis adjacent to one side of the rectangle

solid of revolution

solid resulting from a region in a plane revolved around a line

axis of revolution

the line that the area of a region is resolved around

cross sections taken perpendicular to x-axis

v = ∫ A(x) dx

use a graphing calculator to find the actual position, at 10 seconds, of an object moving in a straight line with velocity v(t) = √(3t - 2) meters per second. at t = 5, the object's distance from the starting point was 12 meters in the positive direction.

~34.509 meters

The graphs of y = 2x and y = 3x² - x³ intersect at x = 0, x = 1, and x = 2. What integral gives the sum of the areas bounded by the graphs?

∫ (0 to 1) (2x - (3x² - x³)) dx + ∫ (1 to 2) ((3x² - x³) - 2x) dx

What integral setup gives the sum of the areas bounded by the graphs of y = - 2/π x + 1 and y = cos x ?

∫ (0 to π/2) (cos x - (- 2/π x + 1)) dx + ∫ (π/2 to π) ((- 2/π x + 1) - cos x) dx

How would you calculate the total distance traveled by a particle on [0, π], given that the velocity is v(t) = - sin( t - π/4 ) ?

∫ (0 to π/4) v(t) dt - ∫ (π/4 to π) v(t) dt

Let R be the region between the graph of y = arctan x, the x-axis, and x = 1.5. What integral gives the area of region R?

∫ (1.5 - tan y) dy, from 0 to arctan(1.5)

Net Change Theorem

∫ F ' (x) = F(b) - F(a)

actual position (given velocity)

∫ v(t) dt + c

displacement on [a, b] (given velocity)

∫ v(t) dt = sum of areas (watch signs)

total distance traveled on [a, b] (given velocity)

∫ |v(t)| dt = sum of absolute value of areas

area of square

A = b^2

area of semicircle

A = π/8 b^2

A particle moves along the x-axis with velocity given by v(t) = (t - 1) e^(1-t) for time t >= 0. If the particle is at position x = 3 at time t = 0, which integral setup gives the position of the particle at time t = 2?

3 + ∫ (t-1) * e^(1-t) dt from 0 to 2

What is the area of the regions enclosed by y = x / 2 and y = sin² x ?

0.249

What is the area of the region in the first quadrant bounded on the left by the graph of x = y√(y⁴ + 1) and on the right of the graph of x = 2y ?

0.537

For time t >= 0, the acceleration of an object moving in a straight line is given by a(t) = sin (t² / 3). What is the net change in velocity from time t = 0.75 to time t = 2.25?

0.984

An electric monorail transport is slowing down at a rate of 2t + 25 ft per second per second, where t is the time in seconds. by how many feet per second does the monorail slow down between t = 1 and t = 5 seconds?

124 ft/sec

what is the total distance traveled by a particle moving along the x-axis with velocity v(t) = -t^2 + 7t - 10 meters per second on the time interval [1, 7]?

15 meters

Given f(x) = 3x^2 - 2x, find the average value of f on the interval [1, 4]

16

The base of a solid is the triangular region in the first quadrant bounded by the graph of y = 5 - 5/3 x and the x- and y-axes. For the solid, each cross section perpendicular to the x-axis is a square. What is the volume of the solid?

25

The number of liters in oil in a tank is changing at a rate r(t) = 0.26t liters per minute, where t is the time in minutes. Initially, the tank contained 14 liters of oil. How much oil is left in the tank after 10 minutes?

27 liters

The base of a solid is the region in the first quadrant bounded by the x- and y-axes and the graph of y = (x-2)² / [2 (x+1)]. For the solid, each cross section perpendicular to the x-axis is a rectangle whose height is three times its width in the xy-plane. What is the volume of this solid?

3.012

find the volume of the solid whose base is a region bounded by the circle x^2 + y^2 = 9 with the cross sections taken perpendicular to the x-axis being isosceles right triangles with the hypotenuse lying on the base of the circle

36

Let R be the region in the first quadrant bounded by the graphs of x = y³ and x = 4y. What is the area of R?

4

The rate at which sand is poured into a bag is modeled by the function r given by r(t) = 15t - 2t², where r(t) is measured in milliliters per second and t is measured in seconds after the sand begins pouring. How many milliliters of sand accumulate in the bag from time t = 0 to time t = 2 ?

74 / 3

Let R be the region in the first quadrant bounded by the graphs of y = 1 - x³ and y = 1 - x. The region R is the base of a solid. For the solid, each cross section perpendicular to the x-axis is a square. What is the volume of the solid?

8 / 105

Given f(x) = 3x^2 - 2x, find the value(s) of c guaranteed by the Mean Value Theorem for Integrals

8/3

calculate the area between the two curves f(x) = 14x + x^2 - 2x^3 g(x) = x^2 - 4x

81

find the volume of the region between y = 4 - x^2 y = 0 when revolved about the line y = -1

832π / 15

If the average value of the function f on the interval [1, 4] is 8, what is the value of ∫ (3 f(x) + 2x) dx from 1 to 4 ?

87

The graph of a particle's velocity, v(t) at time t is positive twice and negative once on [0,8]. Over this time, the particle's displacement is 100 units to the right and the particle travels a total distance of 1875. What is the total distance the particle travels while moving to the left?

887.5

find the volume of y = x^2 revolved about y-axis from [0, 4]

8π

area of equilateral triangle

A = (√(3) / 4) b^2

area of isosceles right triangle with base as leg

A = 1/2 b^2

area of rectangle

A = 1/2 b^2

area of isosceles right triangle with hypotenuse as leg

A = 1/4 b^2

Mean Value Theorem

If f is continuous on the closed interval [a, b], then there exists a number c in the closed interval [a, b] such that ∫ f(x) dx = f(c) (b - a)

arc length (formula)

L = ∫ √( 1 + (f '(x))^2 ) dx

arc length (definition)

L on the interval [a, b], where (a, f(a)) and (b, f(b)) are endpoints

volume of a cone

V = 1/3 π r^2 h

washer method formula

V = π ∫ [ (R(x))^2 - (r(x))^2 ] dx (R is the function with the greatest distance from the axis of revolution)

disc method formula

V = π ∫ [R(x)]^2 dx

cross sections taken perpendicular to y-axis

V = ∫ A(y) dy

smooth function

a function f is smooth on an interval if it has a derivative f' that is continuous throughout the interval.