Properties of Definite Integrals

Front

What is the property of definite integrals regarding the sum of integrals over adjacent intervals?

Back

If \( a < b < c \), then \( \int_a^c f(x)dx = \int_a^b f(x)dx + \int_b^c f(x)dx \).

Front

How do you evaluate the definite integral \( \int_a^b f(x)dx \) if you know \( \int_a^c f(x)dx \) and \( \int_c^b f(x)dx \)?

Back

You can use the property of additivity: \( \int_a^b f(x)dx = \int_a^c f(x)dx + \int_c^b f(x)dx \).

Front

What does it mean if a definite integral evaluates to a negative value?

Back

It indicates that the area under the curve of the function \( f(x) \) from \( a \) to \( b \) is below the x-axis, resulting in a net negative area.

Front

What is the relationship between the definite integral and the area under the curve?

Back

The definite integral \( \int_a^b f(x)dx \) represents the net area between the curve \( f(x) \) and the x-axis from \( x=a \) to \( x=b \).

Front

If \( f(x) \) is an odd function, what can be said about the definite integral over a symmetric interval?

Back

If \( f(x) \) is odd, then \( \int_{-a}^a f(x)dx = 0 \).

Front

What is the effect of reversing the limits of integration on a definite integral?

Back

Reversing the limits of integration changes the sign: \( \int_a^b f(x)dx = -\int_b^a f(x)dx \).

Front

How can you determine the value of an integral if you have the values of integrals over subintervals?

Back

You can use the property of additivity to combine the values of the integrals over the subintervals.

Front

What is the Fundamental Theorem of Calculus in relation to definite integrals?

Back

The Fundamental Theorem of Calculus states that if \( F \) is an antiderivative of \( f \), then \( \int_a^b f(x)dx = F(b) - F(a) \).

Front

What is the significance of the area above and below the x-axis in a definite integral?

Back

The area above the x-axis contributes positively to the integral, while the area below the x-axis contributes negatively.

Front

How do you evaluate \( \int_a^b f(x)dx \) if \( f(x) \) is piecewise defined?

Back

Evaluate the integral over each piece separately and then sum the results.