Using additivity over intervals to find a subinterval integral

Since $2 < 4 < 7$, the additivity property of definite integrals gives $\int_2^7 h(x)\,dx = \int_2^4 h(x)\,dx + \int_4^7 h(x)\,dx.$ This lets us split the integral over $[2,7]$ into the two adjacent pieces $[2,4]$ and $[4,7]$.

Substituting the known values $10 = 4 + \int_4^7 h(x)\,dx$, so solving for the unknown integral gives $\int_4^7 h(x)\,dx = 10 - 4 = 6$.

Reversing the bounds of a definite integral negates its value. Since the integral from four to seven of h equals six, the integral from seven to four of h equals negative six. This follows directly from the sign-reversal property of definite integrals.