Integral

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 continuous on [a, b], then: 1) If F is an antiderivative of f on [a, b], then \( \int_a^b f(x)dx = F(b) - F(a) \).

Front

Define the definite integral.

Back

A definite integral is an integral with upper and lower limits, representing the signed area under the curve of a function between two points. It is denoted as \( \int_a^b f(x)dx \).

Front

What is the purpose of Riemann sums?

Back

Riemann sums are used to approximate the area under a curve by dividing the region into rectangles and summing their areas. They form the basis for defining the definite integral.

Front

Explain the difference between left and right Riemann sums.

Back

Left Riemann sum uses the left endpoints of subintervals to determine the height of rectangles, while right Riemann sum uses the right endpoints.

Front

What is the trapezoidal rule?

Back

The trapezoidal rule approximates the area under a curve by dividing it into trapezoids instead of rectangles, providing a more accurate estimate than Riemann sums.

Front

How do you evaluate \( \int_0^7 f'(x)dx \)?

Back

By the Fundamental Theorem of Calculus, \( \int_0^7 f'(x)dx = f(7) - f(0) \).

Front

What is an antiderivative?

Back

An antiderivative of a function f is a function F such that \( F' = f \). It represents the reverse process of differentiation.

Front

Calculate \( \int_1^4 \left(\frac{1}{x^2} - \sqrt{x}\right)dx \).

Back

The result is \( -\frac{47}{12} \).

Front

What is the significance of continuity in integration?

Back

Continuity of a function on a closed interval ensures that the function can be integrated over that interval, allowing the application of the Fundamental Theorem of Calculus.

Front

Define the term 'subinterval' in the context of integration.

Back

A subinterval is a smaller segment of the interval over which integration is performed, used to approximate the area under the curve.