Properties of Definite Integrals

Front

What is the property of definite integrals that allows the subtraction of two integrals with the same upper limit?

Back

If \( \int_a^b f(x)dx - \int_c^b f(x)dx = \int_a^c f(x)dx \), where \( a < c < b \).

Front

What does the symbol \( \int_a^b f(x)dx \) represent?

Back

The area under the curve of \( f(x) \) from \( x = a \) to \( x = b \) on the x-axis.

Front

If \( \int_{-4}^3 f(x)dx = 9 \) and \( \int_3^5 f(x)dx = -11 \), what is \( \int_{-4}^5 f(x)dx \)?

Back

\( \int_{-4}^5 f(x)dx = \int_{-4}^3 f(x)dx + \int_3^5 f(x)dx = 9 + (-11) = -2 \).

Front

What is the relationship between the integral of a function and its area?

Back

The definite integral of a function gives the net area between the curve and the x-axis, accounting for areas above the x-axis as positive and below as negative.

Front

How can 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

Use the property: \( \int_a^b f(x)dx = \int_a^c f(x)dx + \int_c^b f(x)dx \).

Front

What is the effect of reversing the limits of integration?

Back

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

Front

If \( \int_a^b f(x)dx = 0 \), what can be inferred about the function?

Back

The net area under the curve from \( a \) to \( b \) is zero, meaning the areas above and below the x-axis are equal.

Front

What is the integral of a constant function \( c \) over the interval \( [a, b] \)?

Back

\( \int_a^b c \, dx = c(b - a) \).

Front

How do you interpret \( \int_a^b f(x)dx \) geometrically?

Back

It represents the total area between the curve \( f(x) \) and the x-axis from \( x = a \) to \( x = b \).

Front

What is the Fundamental Theorem of Calculus?

Back

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