Finding a type 1 error Suppose that $T_1,…,T_{10}$ are iid $Exp(λ)$ and the goal is to test if $H_0:λ≤1$ versus $H_a:λ≥2$. Suppose that the test statistic is $S=\sum_{i=1}^{10}T_i$, and rejection of the null occurs when $S≤7$.

I realize that $T_1...T_{10}$ is a gamma distribution  of $S =Gamma(10, \lambda$),
but I am not sure how to use this information to find the p-value for the type 1 error.
 A: You reject when $S \le 7$ where, under $H_0; \lambda \le 1,$
$S \sim \mathsf{Gamma}(10, 1),$ so the probability of Type I Error
(rejecting $H_0$ when $H_0$ is true) is
$\alpha = P(S \le 7\, |\, H_0) = 0.1695.$ [Computation in R, where
pgamma is a gamma CDF.]
pgamma(7, 10, 1)
[1] 0.1695041

Then $\beta = P(x \ge 7\, |\, H_a) = 1 - P(S \le 7\, |\, H_a) =  0.109,$
whee $H_a: \lambda \ge 2.$
1 - pgamma(7, 10, 2)
[1] 0.1093994

In the figure below $\alpha$ is the area under the blue density curve to the
left of $s = 7$ and $\beta$ is the area under the brown density curve to the right
of $s=7.$

R code for figure:
hdr = "Density functions of GAMMA(10,1) [null, blue] and GAMMA(10,2)"
curve(dgamma(x, 10, 1), 0, 20, ylim=c(0,.3), col="blue", lwd=2,
      ylab="Density", xlab="s", main=hdr)
  curve(dgamma(x, 10, 2), add=T, col="brown", lwd=2)
  abline(v=0, col="green2");  abline(h=0, col="green2")
  abline(v=7, col="red", lwd=2, lty="dashed")

A: When computing the Type I error, you assume $H_0$ is true.  So you are finding the $P(S \le 7 | \lambda \le 1)$.  As $\lambda$ gets smaller, we expect $S$ to get larger, so $P(S \le 7 | \lambda \le 1)$ will be the largest when $\lambda = 1$.  $P(S \le 7 | \lambda \le 1) \le P(S \le 7 | \lambda = 1) = P(S \sim Gamma(10, 1) \le 7) = 0.1695$
One other note, the null and alternate hypothesis normally divide the parameter space without holes.  Did you mean to say $H_a: \lambda > 1$?  You can compute this in R with pgamma(7, 10, 1)
Nsims <- 10000

lambda0 <- 1
n <- 10
S <- numeric(Nsims)
for (i in 1:Nsims) {
  Tn <- rexp(n, rate = lambda0) # lambda is E(exp(lambda)) = 1 / lambda
  S[i] <- sum(Tn)
}

hist(S, freq = FALSE)
lines(seq(0,30,length=1000), dgamma(seq(0,30,length=1000), n, lambda0))


#P(S < 7 | lambda0 = 1)

length(which(S <= 7)) / length(S)
#> [1] 0.1723
pgamma(7, 10, 1)
#> [1] 0.1695041

