Properties of Definite Integrals

Front

What is the property of definite integrals that allows the sum of two integrals over adjacent intervals to be combined into a single integral?

Back

The property states that \( \int_a^b f(x)dx + \int_b^c f(x)dx = \int_a^c f(x)dx \).

Front

If \( \int_a^b f(x)dx = C \), what is the value of \( \int_a^b kf(x)dx \) for a constant \( k \)?

Back

The value is \( kC \).

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

Back

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

Front

What is the relationship between the definite integral of a function and its average value over an interval \([a, b]"?

Back

The average value of a function \( f \) over \([a, b]\) is given by \( \frac{1}{b-a} \int_a^b f(x)dx \).

Tags

CCSS.8.F.B.4

CCSS.HSF.IF.B.6

Front

If \( f(x) \) is an even function, what can be said about the definite integral from \(-a\) to \(a\)?

Back

If \( f(x) \) is even, then \( \int_{-a}^a f(x)dx = 2 \int_0^a f(x)dx \).

Front

If \( f(x) \) is an odd function, what can be said about the definite integral from \(-a\) to \(a\)?

Back

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

Front

What is the value of \( \int_a^a f(x)dx \) for any function \( f \)?

Back

The value is 0, since the integral over an interval of zero length is zero.

Front

How can you express the definite integral of a function from \( a \) to \( b \) in terms of its antiderivative \( F(x) \)?

Back

By the Fundamental Theorem of Calculus: \( \int_a^b f(x)dx = F(b) - F(a) \), where \( F' = f \).

Front

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

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 \( x=b \).