Integration

Front

What is the definition of integration in calculus?

Back

Integration is the process of finding the integral of a function, which represents the area under the curve of that function on a given interval.

Front

What is the Fundamental Theorem of Calculus?

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 and summing their areas.

Front

What is the difference between left and right Riemann sums?

Back

Left Riemann sums use the left endpoints of subintervals to determine the height of rectangles, while right Riemann sums use the right endpoints.

Front

How do you calculate the area under a curve using integration?

Back

To calculate the area under a curve, you evaluate the definite integral of the function over the desired interval.

Front

What does it mean for a function to be strictly decreasing?

Back

A function is strictly decreasing if, for any two points x1 and x2 in its domain, if x1 < x2, then f(x1) > f(x2).

Front

What is an antiderivative?

Back

An antiderivative of a function f is a function F such that F' = f, meaning that F is the integral of f.

Front

What is the significance of the area under the curve in real-world applications?

Back

The area under the curve can represent quantities such as total distance traveled, total accumulated value, or total growth over time.

Front

What is the relationship between the shape of a function and the estimation of area using Riemann sums?

Back

The shape of the function affects whether Riemann sums overestimate or underestimate the area; for increasing functions, left sums underestimate and right sums overestimate, while for decreasing functions, the opposite is true.

Front

What is the formula for calculating the area using a left Riemann sum?

Back

The area using a left Riemann sum is calculated as: A ≈ Σ f(x_i) * Δx, where x_i are the left endpoints of the subintervals.