Definite Integrals

Front

What is a definite integral?

Back

A definite integral is a mathematical concept that represents the signed area under a curve defined by a function over a specific interval [a, b]. It is denoted as ∫_a^b f(x) dx.

Front

What does the Fundamental Theorem of Calculus state?

Back

The Fundamental Theorem of Calculus links the concept of differentiation and integration, stating that if F is an antiderivative of f on an interval [a, b], then ∫_a^b f(x) dx = F(b) - F(a).

Front

What is a Riemann Sum?

Back

A Riemann Sum is a method for approximating the total area under a curve by dividing the area into rectangles, calculating the area of each rectangle, and summing them up.

Front

What is the difference between a left Riemann Sum and a right Riemann Sum?

Back

A left Riemann Sum uses the left endpoints of subintervals to determine the height of rectangles, while a right Riemann Sum uses the right endpoints.

Front

For a strictly decreasing function, what type of estimate does a right Riemann Sum provide?

Back

A right Riemann Sum provides an underestimate for a strictly decreasing function.

Front

How do you calculate the area under a curve using definite integrals?

Back

To calculate the area under a curve using definite integrals, evaluate the integral of the function over the specified interval [a, b].

Front

What is the area under the curve y = 5x^4 + 3x + 7 from x = 0 to x = 4?

Back

The area under the curve is 1076.

Front

What is the area under the curve y = 3x^2 - 2x from x = 1 to x = 5?

Back

The area under the curve is 100.

Front

What is the formula for calculating the distance traveled using a left sum?

Back

The distance traveled can be approximated by summing the areas of rectangles formed using the left endpoints of subintervals.

Front

What is the result of evaluating the definite integral of a constant function over an interval?

Back

The result is the product of the constant and the length of the interval.