5.2 The Definite Integral

In the previous section, we saw that if f is continuous on an interval [a,b], then the endpoint and midpoint approximations approach a common limit L as N → ∞:

When f(x) ≥ 0, L is the _____ under the graph of f. In a moment, we will state formally that L  is the _____ _______ of f over [a,b]. Before doing so, we introduce more general approximations called ______ _____

Recall that R_N, L_N and M_N use rectangles of equal width Δx, whose heights are the values of ____ at the endpoints or midpoints of the _______. In Riemann sum approximations, we relax these requirements: The rectangles need not have _____ width and height may be any value of ____ within the subinterval

To specify a Riemann sum, we choose a partion and a set of sample points (Define both)

What is the length of i^th subinterval [x_i-1, x_i]?

Define the norm of P

Given P and C, we construct the rectangle height ____ and base _____ over each subinterval [xi-1, xi], as in Fig 2B (pg 38) This rectangle has area _______ if f(c) ≥ 0. If f(ci) < 0, the rectangle extends below the x-axis, and f(ci)Δxi is the _______ of its area

Represent the Riemann sum in summation notation

Keep in mind that RN, LN, and MN are particular examples of _______ _____ in which Δx_i = ______ for all i, and the sample points c_i are either _______ or _______

Note in Fig 2C (pg 238) that as the norm ||P|| tends to zero (meaning that the rectangles get ______), the number of rectangles N tends to _____ and they approximate the area under the graph more closely. Define integrable

Formally, when do we right: 
L = lim_||P||→0 R(f,P,C) = lim_||P||→0  ^NΣ_i=1 f(c_i)Δx_i
Th limit L is called the _____ _____ of f over [a,b]

The notation ∫ f(x)dx was introduced by Leibniz in 1686. The symbol ∫ is an elongated S standing for ________. The differential dx corresponds to the length _____ along the x-axis

Define the Definite Integral

The definit integral is often called, more simply, the integral of f over [a,b]. The process of computing integrals is called _______ and f(x) is called the _______. The endpoints a and b of [a,b] are called the ______ ____ ______. Finally, we remark that any variable may be used as a variable of integration (this is a "dummy" variable). (pg 239)

Keep in mind that a Riemann sum R(f, P, C) is nothing more than an approximation to _____ based on _______, and that ∫_a^b f(x) dx is the number we obtain in the limit as we take ______ and ______ rectangles

Theorem 1: If f is continuous on [a,b], or if f is continuous with at most finitely many jump discontinuities, then f is ______ over [a,b]

When f(x) ≥ 0, then definite integral defines the area under the graph. To interpret the integral when f(x) takes on both positive and negative values, we define the notion of ______ _____, where regions below the x-axis are considered to have "______ area" Fig 4 (pg 230)

Signed area of a region formula:

Now observe that a Riemann sum is equal to the signed area of the corresponding rectangles:
R(f, C, P) = ?

Indeed, if f(ci) < 0, then the corresponding rectangle lies below the x-axis and has signed area _________ Fig 5 (pg 240). What is the limit of the Riemann sums?

∫_a^b f(x)dx = ?

In the rest of this section, we discuss some basic properties of definite integrals. First, we note that the integral of a constant function f(x) = C over [a,b] is the signed area _______ of rectangle as in Fig 7 (pg 240)

State Integral of a Constant Theorem (2)

State the Linearity of the Definite Integral Theorem

∫₀^b x²dx = ?

Reversing the the Limits of Integration (Definition)

When a = b, the interval [a,b] = [a,a]has length zero and we define the definite integral to be _______

Definite integrals satisfy an important additivity property: If f is integrable and a ≤ b ≤ c as in Fig 9 (pg 242), then the integral from a to c is equal to the integral from:

State the Additivity for Adjacent Intervals Theorem

State the Comparison Theorem