What does the Fundamental Theorem of Calculus (FTC) link together?
Derivative and area
The Power Rule for integration states that ∫xndx=n+1xn+1+C, where n= -1
The Constant Multiple Rule states that ∫k⋅f(x)dx=k∫f(x)dx
What does a definite integral represent geometrically?
Area under a curve
Definite integrals are evaluated using the formula ∫abf(x)dx=F(b)− F(a)
Steps to find the area under f(x) = x^{2}</latex> from a=0 to b=2
1️⃣ Find the antiderivative: F(x)=3x3
2️⃣ Evaluate at the limits: F(2)=38, F(0)=0
3️⃣ Subtract: F(2)−F(0)=38
If F(x)=∫axf(t)dt, then F′(x)=f(x)
What is Part 2 of the Fundamental Theorem of Calculus used to evaluate?
Definite integrals
The Power Rule states that \int x^{n} dx = \frac{x^{n + 1}}{n + 1} +</latex> C
If F(x)=∫axf(t)dt, then F′(x)=f(x)
The Fundamental Theorem of Calculus consists of two key parts
What does the first part of the Fundamental Theorem of Calculus state about the derivative of \int_{a}^{x} f(t) dt</latex>?
F′(x)=f(x)
F(x)=∫0xt2dt=3x3
If f(x) is continuous on [a,b] and F(x) is any antiderivative of f(x), then ∫abf(x)dx=F(b)−F(a) is the statement of the Fundamental Theorem of Calculus, Part 2.
Evaluate ∫02xdx using the Fundamental Theorem of Calculus.
2
Steps to evaluate definite integrals using the Fundamental Theorem of Calculus
1️⃣ Find the antiderivative F(x) of f(x)
2️⃣ Evaluate F(b)−F(a), where a and b are the limits of integration
What is the antiderivative of x2?
3x3
The value of ∫13x2dx is \frac{26}{3}
The Fundamental Theorem of Calculus Part 2 requires finding an antiderivative to evaluate a definite integral.
Match the concept with its description:
Antiderivative ↔️ A function whose derivative equals the original function
FTC Part 2 ↔️ Method to evaluate definite integrals using antiderivatives
What is the antiderivative of x?
2x2
The value of \int_{0}^{2} x dx</latex> is 2
Steps to apply the Power Rule for integration
1️⃣ Add 1 to the exponent of x
2️⃣ Divide by the new exponent
3️⃣ Add the constant of integration C
The Constant Multiple Rule states that ∫k⋅f(x)dx=k∫f(x)dx.
What does a definite integral represent geometrically?
Area under a curve
The formula ∫abf(x)dx=F(b)−F(a) is used in the Fundamental Theorem of Calculus to evaluate definite integrals.
Match the concept with its description:
Definite Integral ↔️ Area under a curve between two limits
FTC ↔️ Links derivatives to integrals
What is the derivative of ∫axf(t)dt?
f(x)</latex>
If f(x) is continuous and F(x) is an antiderivative of f(x), then ∫abf(x)dx=F(b)−F(a) is the Fundamental Theorem of Calculus, Part 2.
What is the antiderivative of x^{2}</latex>?
3x3
An antiderivative of f(x) is a function whose derivative equals f(x).
Match the concept with its formula:
Antiderivative ↔️ F'(x) = f(x)</latex>
FTC Part 2 ↔️ ∫abf(x)dx=F(b)−F(a)
The value of ∫02xdx is 2
What is the formula for finding the area between two curves f(x) and g(x)?
∫ab(f(x)−g(x))dx
The area between f(x)=x2 and g(x)=x from 0 to 1 is 61.