Math Revision Flashcards

Flashcards for reviewing applications of integrals, including volumes of solids of revolution and integration techniques.

What are the 5Es of the lesson?

Engage, Explore, Explain, Elaborate, Evaluate

What is the formula for the Disk Method around the x-axis?

V = π ∫ [f(x)]² dx from a to b

In the Washer Method, what do R(x) and r(x) represent?

R(x) = outer radius, r(x) = inner radius

What is the formula for the Washer Method around the x-axis?

V = π ∫ (R(x)² – r(x)²) dx from a to b

What is the Shell Method formula revolving around the y-axis?

V = 2π ∫ x * f(x) dx from a to b

What is the formula for Arc Length?

L = ∫ √(1 + [f'(x)]²) dx from a to b

What is the formula for the Surface Area of Revolution about the x-axis?

S = 2π ∫ f(x)√(1 + [f'(x)]²) dx from a to b

What is the Integration by Parts Formula?

∫ u dv = uv - ∫ v du

What does LIATE stand for in the LIATE rule?

Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential

What is the formula for the Disk Method when rotating around the x-axis?

V = π ∫ [f(x)]² dx from a to b

What is the volume of the solid formed when the region bounded by the curve y = √x, and the lines x = 0 and x = 4 is revolved about the x-axis?

8π

Which region (y = x or y = x²) would give a greater volume when rotated about the x-axis over the interval [0, 1], and why?

y = x generates a greater volume because 1/3 > 1/5

What is the formula for the Washer Method around the x-axis?

V = π ∫ ([R(x)]² – [r(x)]²) dx from a to b

In the context of volume calculation using the washer method, what does R(x) represent?

Outer radius

In the context of volume calculation using the washer method, what does r(x) represent?

Inner radius

What is the volume of the solid formed when the region between y = √x and y = x from x = 0 to 1 is revolved around the x-axis?

π/10

What is the Shell Method Formula for One Function, when rotating around the y-axis?

V = 2π ∫ x * f(x) dx

What is the volume of the solid formed when the region between the curve y=x^2 from x=0 to x=2 is revolved around the y-axis?

8π

What is the Shell Method Formula with Two Functions when rotating around the y-axis?

V = 2π ∫ x * (f(x) - g(x)) dx

What is the volume of the solid formed when the region between y = √x and y = x from x = 0 to x = 1 is revolved around the y-axis?

2π/15

What is the Surface Area of a Solid of Revolution (around the x-axis)?

S = 2π ∫ f(x)√(1 + [f'(x)]²) dx

What is the surface area of the solid generated by rotating the line y = 3 about the x-axis between x = 1 and x = 4?

18π

What is the formula for integration by parts?

∫ u dv = uv - ∫ v du

What does LIATE stand for?

L: Logarithmic, I: Inverse trigonometric, A: Algebraic, T: Trigonometric, E: Exponential

Solve the following integral using integration by parts: ∫ x ln(x) dx

(x²/2)ln(x) - x²/4 + C

Solve the integral: ∫ x*e^x dx

x*e^x - e^x + C

Solve ∫ x cos x dx

x sin x + cos x + C

Calculate the total energy released by the material from the start of the experiment (t=0) to 3 hours after the start (t=3) given P(t) = t*e^(-t)

1 - 4/e^3

If f(x) = 2x - 1 and g(x) = x + 4, what is the value of f(g(3))?

g(3) = 3 + 4 = 7; f(g(3)) = f(7) = 2(7) - 1 = 13

A recipe requires 5 cups of milk for every 3 cups of oats. How many cups of milk are needed if you use 15 cups of oats?

x = 25

Solve the system of equations: 3x + 2y = 16 and x - y = 2. What is the value of y?

y = 2

The equation of a circle is given by: (x + 2)² + (y – 5)² = 36. What is the radius of the circle?

r = 6

Find the volume generated when the region under y = 3x from x = 0 to x = 2 is rotated about the y-axis (DIFFERENTIATED ACTIVITIES)

Volume generated when rotating y=3x from x=0 to x=2 about y-axis

Use the shell method to find the volume formed by rotating the region under y = x² from x = 0 to x = 1 about the y-axis (DIFFERENTIATED ACTIVITIES)

Volume with the shell method when rotating y=x^2, x = [0,1]

In heat exchangers, why is shell thickness important in discussion based answer (DIFFERENTIATED ACTIVITIES)

Heat transfer accross cylindrical walls

Rotate region between y = x and y = x² (from x = 0 to x = 1) about the y-axis using the shell method (DIFFERENTIATED ACTIVITIES)

Rotating region between y=x and y = x^2(D)

Find the volume of the solid obtained by rotating the region bounded by y = √x, x = 1, and the x-axis about the line x = 2 (DIFFERENTIATED ACTIVITIES)

Rotating region between y, x and x about line: x=2

How can we use the shell method for rotating function around y-axis?

Integration application with shell method for rotating function around y-axis