M459 Integrals Review

Front

What is the definition of 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 specified interval [a, b]. It is denoted as ∫_a^b f(x) dx.

Front

What is the Fundamental Theorem of Calculus?

Back

The Fundamental Theorem of Calculus states 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 region into rectangles, calculating the area of each rectangle, and summing these areas.

Front

What is the difference between left and right Riemann sums?

Back

Left Riemann sums use the left endpoints of sub-intervals 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 a left Riemann sum?

Back

To calculate the area under a curve using a left Riemann sum, divide the interval into n sub-intervals, determine the height of the function at the left endpoint of each sub-interval, multiply by the width of the sub-intervals, and sum the areas of the rectangles.

Front

What is the formula for the area of a rectangle?

Back

The area of a rectangle is calculated as A = width × height.

Front

What is the significance of the limits of integration in a definite integral?

Back

The limits of integration, a and b, define the interval over which the area under the curve is calculated in a definite integral.

Front

What is the relationship between integration and differentiation?

Back

Integration and differentiation are inverse processes; the derivative of an integral function gives the original function, and the integral of a derivative returns the original function (up to a constant).

Front

What is the trapezoidal rule?

Back

The trapezoidal rule is a numerical method for estimating the value of a definite integral by approximating the area under the curve as a series of trapezoids.

Front

How do you evaluate a definite integral using substitution?

Back

To evaluate a definite integral using substitution, choose a substitution that simplifies the integral, change the limits of integration accordingly, and then integrate the new function.