(g) One … in Figure 5.10 gives the exact area

True or false, with reason or example.

True or false, with reason or example.

(a) Every continuous $v(x)$ has an antiderivative $f(x).$

(b) If $v(x)$ is not continuous, $S$ and $s$ approach different limits.

(c) If $S$ and $s$ approach $A$ as $\Delta x\to 0,$ then all Riemann sums $S^{\ast }$ in equation (9) also approach $A.$

(d) If $v_1(x)+v_2(x)=v_3(x),$ their upper sums satisfy $S_1+S_2=S_3.$

(e) If $v_1(x)+v_2(x)=v_3(x),$ their Riemann sums at the midpoints $x_k^{\ast }$ satisfy $S_1^{\ast }+S_2^{\ast }=S_3^{\ast }.$

(f) The midpoint sum is the average of $S$ and $s.$

(g) One $x_k^{\ast }$ in Figure 5.10 gives the exact area.

Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) · §5.5 The Definite Integral, #33