Definite Integrals

These flashcards cover key concepts related to definite integrals, including definitions, applications, and examples.

What is a definite integral?

The area under y = f(x) between x = a & x = b.

What do a and b represent in a definite integral?

a is the lower bound/limit and b is the upper bound/limit.

How are definite integrals approximated?

Using rectangles to approximate the area under the curve.

What is the form of the approximating sums for definite integrals?

f(x₁)Δx + f(x₂)Δx + … + f(xn)Δx = Σ f(x)Δx.

What is the difference between lower sum and upper sum in definite integrals?

Lower sum is usually an underestimate, while upper sum is an overestimate of the area.

What happens to approximating sums with larger values of n?

They give better approximations of the definite integral.

In rectangular approximation of integrals, what do the sample points refer to?

The x values where you measure the height of the rectangle.

If f(1) = 3, f(2) = 4, f(3) = 3, and f(4) = 2, what would be the left sum?

Left sum: f(1) + f(2) + f(3) = 3 + 4 + 3 = 10.

If f(1) = 3, f(2) = 4, f(3) = 3, and f(4) = 2, what would be the right sum?

Right sum: f(2) + f(3) + f(4) = 4 + 3 + 2 = 9.

In the context of definite integrals, what does the symbol Δx represent?

The width of the rectangles.