- Evaluate these integrals. (Some of these are indefinite integrals while some are definite integrals. Recall what it means to “evaluate” an indefinite integral versus a definite integral, i.e. what should the final result be?) \[ \int\limits x^{10} + 10^x + \frac{10}{x} + \frac{x}{10} + \sqrt[10]{x} + \sqrt[x]{10} + 10^{10} \,\mathrm{d}x \qquad \int\limits 3-7\cos(x) \,\mathrm{d}x \qquad \int\limits_{1}^{2} x^7 \,\mathrm{d}x \qquad \int\limits_{0}^{4} \bigl| 3-2x \bigr|-1 \,\mathrm{d}x \]
- Suppose that \(\int\limits_{3}^{7} f(x)\,\mathrm{d}x = 8\) for some function \(f.\) What must the value of the following integrals be? \[ \int\limits_{3}^{3} f(x) \,\mathrm{d}x \qquad \qquad \qquad \int\limits_{7}^{3} f(x) \,\mathrm{d}x \qquad \qquad \qquad \int\limits_{7}^{3} 6-2f(x) \,\mathrm{d}x \]
- A car is driving eastward on I-70 Freeway away from Grand Junction. Its speed, in miles-per-hour, is given by the function \({f\left(t\right)=\frac{1}{25}t^{2}-\frac{1}{5}t+65}\) where \(t\) is the number of hours since it left GJ. How far does it travel in the first four hours? What is the average speed of the car in those first four hours?
- What is the area of the region bound between the parabola with equation \(y = x^2-10x+21\) and the \(x\)-axis?
- What is the total (non-signed) area of the region bound between the graph of \(f(x) = x^3-3x^2+2x\) and the \(x\)-axis?
- What is the area of the region inside the parabola given by the equation \(y = x^2\) below the line \(y = 5\,?\)